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In this paper we introduce a natural model for the realization space of a polytope up to projective equivalence which we call the slack realization space of the polytope. The model arises from the positive part of an algebraic variety determined by the slack ideal of the polytope. This is a saturated determinantal ideal that encodes the combinatorics of the polytope. We also derive a new model of the realization space of a polytope from the positive part of the variety of a related ideal. The slack ideal offers an effective computational framework for several classical questions about polytopes such as rational realizability, non-prescribability of faces, and realizability of combinatorial polytopes., Comment: The original arXiv submission of this paper has now been split into two parts; this paper describing the slack realization space of a polytope and applications of the slack ideal, and a second paper focussing on toric slack ideals and projective uniqueness

Two permutations of $[n]=\{1,2 \ldots n\}$ are \textit{$k$-neighbor separated} if there are two elements that are neighbors in one of the permutations and that are separated by exactly $k-2$ other elements in the other permutation. Let the maximal number of pairwise $k$-neighbor separated permutations of $[n]$ be denoted by $P(n,k)$. In a previous paper, the authors have determined $P(n,3)$ for every $n$, answering a question of K\"orner, Messuti and Simonyi affirmatively. In this paper we prove that for every fixed positive integer $\ell $, $$P(n,2^\ell+1) = 2^{n-o(n)}. $$ We conjecture that for every fixed even $k$, $P(n,k)=2^{n-o(n)}$. We also show that this conjecture is asymptotically true in the following sense $$\lim_{k \rightarrow \infty} \lim_{n \rightarrow \infty} \sqrt[n]{P(n,k)}=2.$$ Finally, we show that for even $n$, $P(n,n)= 3n/2$.

In this paper, we study zero-one laws for the Erd\H{o}s--R\'{e}nyi random graph model $G(n,p)$ in the case when $p = n^{-\alpha}$ for $\alpha>0$. For a given class $\mathcal{K}$ of logical sentences about graphs and a given function $p=p(n)$, we say that $G(n,p)$ obeys the zero-one law (w.r.t. the class $\mathcal{K}$) if each sentence $\varphi\in\mathcal{K}$ either a.a.s. true or a.a.s. false for $G(n,p)$. In this paper, we consider first order properties and monadic second order properties of bounded \textit{quantifier depth} $k$, that is, the length of the longest chain of nested quantifiers in the formula expressing the property. Zero-one laws for properties of quantifier depth $k$ we call the \textit{zero-one $k$-laws}. The main results of this paper concern the zero-one $k$-laws for monadic second order properties (MSO properties). We determine all values $\alpha>0$, for which the zero-one $3$-law for MSO properties does not hold. We also show that, in contrast to the case of the $3$-law, there are infinitely many values of $\alpha$ for which the zero-one $4$-law for MSO properties does not hold. To this end, we analyze the evolution of certain properties of $G(n,p)$ that may be of independent interest.

In this paper, lower bounds on the probability of a finite union of events are considered, i.e. $P\left(\bigcup_{i=1}^N A_i\right)$, in terms of the individual event probabilities $\{P(A_i), i=1,\ldots,N\}$ and the sums of the pairwise event probabilities, i.e., $\{\sum_{j:j\neq i} P(A_i\cap A_j), i=1,\ldots,N\}$. The contribution of this paper includes the following: (i) in the class of all lower bounds that are established in terms of only the $P(A_i)$'s and $\sum_{j:j\neq i} P(A_i\cap A_j)$'s, the optimal lower bound is given numerically by solving a linear programming (LP) problem with $N^2-N+1$ variables; (ii) a new analytical lower bound is proposed based on a relaxed LP problem, which is at least as good as the bound due to Kuai, et al.; (iii) numerical examples are provided to illustrate the performance of the bounds., Comment: SIAM Journal on Discrete Mathematics, 2016

A "book with k pages" consists of a straight line (the "spine") and k half-planes (the "pages"), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a k-page book drawing (or simply a k-page drawing). The k-page crossing number nu_k(G) of a graph G is the minimum number of crossings in a k-page drawing of G. In this paper we investigate the k-page crossing numbers of complete graphs K_n. We use semidefinite programming techniques to give improved lower bounds on nu_k(K_n) for various values of k. We also use a maximum satisfiability reformulation to calculate the exact value of nu_k(K_n) for several values of k and n. Finally, we investigate the best construction known for drawing K_n in k pages, calculate the resulting number of crossings, and discuss this upper bound in the light of the new results reported in this paper., pdfLaTeX, 26 pages

In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions $f: A \to \{0,1\}$ and $f: A \to \mathbb{Z}$ having given line sums in certain directions have been analyzed. Here $A$ was a block in $\mathbb{Z}^n$ with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that $A$ is an arbitrary or convex finite set in $\mathbb{Z}^n$. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of he following type: if the line sums in $k$ directions of a function $h: A \to [0,1]$ are known, then there exists a function $f: A \to \{0,1\}$ such that its line sums differ by at most $k$ from the corresponding line sums of $h$., Comment: 16 pages

Let $G$ be a simple, undirected, finite graph with vertex set $V(G)$ and edge set $E(G)$. A $k$-dimensional box is a Cartesian product of closed intervals $[a_1,b_1]\times [a_2,b_2]\times\cdots\times [a_k,b_k]$. The boxicity of $G$, box$(G)$, is the minimum integer $k$ such that $G$ can be represented as the intersection graph of $k$-dimensional boxes; i.e., each vertex is mapped to a $k$-dimensional box and two vertices are adjacent in $G$ if and only if their corresponding boxes intersect. Let $\mathcal{P}=(S,P)$ be a poset, where $S$ is the ground set and $P$ is a reflexive, antisymmetric and transitive binary relation on $S$. The dimension of $\mathcal{P}$, $\dim(\mathcal{P})$, is the minimum integer $t$ such that $P$ can be expressed as the intersection of $t$ total orders. Let $G_{\mathcal{P}}$ be the underlying comparability graph of $\mathcal{P}$; i.e., $S$ is the vertex set and two vertices are adjacent if and only if they are comparable in $\mathcal{P}$. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset $\mathcal{P}$, box$(G_{\mathcal{P}})/(\chi(G_{\mathcal{P}})-1) \le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$, where $\chi(G_{\mathcal{P}})$ is the chromatic number of $G_{\mathcal{P}}$ and $\chi(G_{\mathcal{P}})\ne1$. It immediately follows that if $\mathcal{P}$ is a height-$2$ poset, then box$(G_{\mathcal{P}})\le \dim(\mathcal{P})\le 2\mbox{box}(G_{\mathcal{P}})$ since the underlying comparability graph of a height-$2$ poset is a bipartite graph. The second result of the paper relates the boxicity of a graph $G$ with a natural partial order associated with the extended double cover of $G$, denoted as $G_c$: Note that $G_c$ is a bipartite graph with partite sets $A$ and $B$ which are copies of $V(G)$ such that, corresponding to every $u\in V(G)$, there are two vertices $u_A\in A$ and $u_B\in B$ and $\{u_A,v_B\}$ is an edge in $G_c$ if and only if either $u=v$ or $u$ is adjacent to $v$ in $G$. Let $\mathcal{P}_c$ be the natural height-$2$ poset associated with $G_c$ by making $A$ the set of minimal elements and $B$ the set of maximal elements. We show that $\frac{\mbox{box}(G)}{2} \le \dim(\mathcal{P}_c) \le 2\mbox{box}(G)+4$. These results have some immediate and significant consequences. The upper bound $\dim(\mathcal{P})\le 2\mbox{box}(G_\mathcal{P})$ allows us to derive hitherto unknown upper bounds for poset dimension such as $\dim(\mathcal{P})\le 2\mbox{ tree width }(G_{\mathcal{P}})+4$, since boxicity of any graph is known to be at most its tree width $+\; 2$. In the other direction, using the already known bounds for partial order dimension we get the following: (1) The boxicity of any graph with maximum degree $\Delta$ is $O(\Delta\log^2\Delta)$, which is an improvement over the best-known upper bound of $\Delta^2+2$. (2) There exist graphs with boxicity $\Omega(\Delta\log\Delta)$. This disproves a conjecture that the boxicity of a graph is $O(\Delta)$. (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on $n$ vertices with a factor of $O(n^{0.5-\epsilon})$ for any $\epsilon>0$ unless $NP=ZPP$.

Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R^3 satisfying minimal rigidity constraints (>= 3 contacts per sphere and >= 3n-6 total contacts). We derive such packings for n = 9, (iii) the number of ground states (i.e. packings with the maximal number of contacts) oscillates with respect to n, (iv) for 10, The updates that have been made to version II are as follows: (i) some updates to the main text have been made, primarily correcting the number of packings reported in PRL, 103, 118303 and in version I of this paper for n = 9,10. (ii) The supplemental information for the paper has been included

In this paper we show examples for applications of the Bombieri-Lang conjecture in additive combinatorics, giving bounds on the cardinality of sumsets of squares and higher powers of integers. Using similar methods we give bounds on the sum-product problem for matchings., Comment: 14 pages

Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph $G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$ vertices with chromatic number $k\geq 4$. In this paper, we study the same problem with the additional constraint that $G$ is $\ell$-connected. For $2$-connected graphs $G$, we prove a tight bound \[ P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{n-k}), \] and show that equality is only achieved if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an asymptotically tight upper bound \[ P_G(k) \le k!(k-1)^{n-\ell - k + 1} + O((k-2)^n), \] and provide a matching lower bound construction. For the ranges $k \geq \ell$ or $\ell \geq (k-2)(k-1)+1$ we further find the unique graph maximizing $P_G(k)$. We also consider generalizing $\ell$-connected graphs to connected graphs with minimum degree $\delta$.

Given a large graph H, does the binomial random graph G(n, p) contain a copy of H as an induced subgraph with high probability? This classical question has been studied extensively for various graphs H, going back to the study of the independence number of G(n, p) by Erdos and Bollobas and by Matula in 1976. In this paper we prove an asymptotically best possible result for induced matchings by showing that if C/n

Let $B_n^{(k)}$ be the book graph which consists of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, and let $C_m$ be a cycle of length $m$. In this paper, we first determine the exact value of $r(B_n^{(2)}, C_m)$ for $\frac{8}{9}n+112\le m\le \lceil\frac{3n}{2}\rceil+1$ and $n \geq 1000$. This answers a question of Faudree, Rousseau and Sheehan (Cycle--book Ramsey numbers, {\it Ars Combin.,} {\bf 31} (1991), 239--248) in a stronger form when $m$ and $n$ are large. Building upon this exact result, we are able to determine the asymptotic value of $r(B_n^{(k)}, C_n)$ for each $k \geq 3$. Namely, we prove that for each $k \geq 3$, $r(B_n^{(k)}, C_n)= (k+1+o_k(1))n.$ This extends a result due to Rousseau and Sheehan (A class of Ramsey problems involving trees, {\it J.~London Math.~Soc.,} {\bf 18} (1978), 392--396)., Comment: Journal Ref. SIAM J. Discrete Math

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{\v{r}}{\'\i}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim \emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension., Comment: Parts of this paper have previously been reported in arXiv submission arXiv:1902.08223

In this paper we describe a parallel algorithm for generating all non-isomorphic rank $3$ simple matroids with a given multiplicity vector. We apply our implementation in the HPC version of GAP to generate all rank $3$ simple matroids with at most $14$ atoms and a splitting characteristic polynomial. We have stored the resulting matroids alongside with various useful invariants in a publicly available, ArangoDB-powered database. As a byproduct we show that the smallest divisionally free rank $3$ arrangement which is not inductively free has $14$ hyperplanes and exists in all characteristics distinct from $2$ and $5$. Another database query proves that Terao's freeness conjecture is true for rank $3$ arrangements with $14$ hyperplanes in any characteristic., Improved exposition

The main theme of this paper is the enumeration of the occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given pattern $v$ exactly $r$ times. In addition, we study the asymptotic behavior of the random variable $X_n,$ the number of pattern occurrences in a random $n$-array word. The two topics are closely related through the identity $P(X_n=r) = $ $\frac{1}{k^n}f_r^v(k,n).$ In particular, we show that for any $r\geq 0,$ the Stanley-Wilf sequence $\bigl(f_r^v(k,n)\bigr)^{1/n}$ converges to a limit independent of $r,$ and determine the value of the limit. We then obtain several limit theorems for the distribution of $X_n,$ including a CLT, large deviation estimates, and the exact growth rate of the entropy of $X_n.$ Furthermore, we introduce a concept of weak avoidance and link it to a certain family of non-product measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley-Wilf sequences, to pattern occurrences in permutations., Comment: 33 pages

Let $\Gamma$ be a graph with vertex set $V(\Gamma)$. A subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if $C$ is an independent set of $\Gamma$ and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if there exists a Cayley graph of $G$ which admits $C$ as a perfect code. A group $G$ is said to be code-perfect if every proper subgroup of $G$ is a perfect code of $G$. In this paper we prove that a group is code-perfect if and only if it has no elements of order $4$. We also prove that a proper subgroup $H$ of an abelian group $G$ is a perfect code of $G$ if and only if the Sylow $2$-subgroup of $H$ is a perfect code of the Sylow $2$-subgroup of $G$. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian $2$-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group., Comment: This is the final version to be published in SIAM Journal on Discrete Mathematics, 18 pp

The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon, and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of the triangulation appears exactly $r$ times. This notion of a rainbow cycle extends in a natural way to other flip graphs. In this paper we investigate the existence of $r$-rainbow cycles for three different flip graphs on classes of geometric objects: the aforementioned flip graph of triangulations of a convex $n$-gon, the flip graph of plane trees on an arbitrary set of $n$ points, and the flip graph of non-crossing perfect matchings on a set of $n$ points in convex position. In addition, we consider two flip graphs on classes of non-geometric objects: the flip graph of permutations of $\{1,2,\dots,n\}$ and the flip graph of $k$-element subsets of $\{1,2,\dots,n\}$. In each of the five settings, we prove the existence and non-existence of rainbow cycles for different values of $r$, $n$ and~$k$.

Let $G$ be an $n$-vertex graph with $n\ge 3$. A classic result of Dirac from 1952 asserts that $G$ is hamiltonian if $\delta(G)\ge n/2$. Dirac's theorem is one of the most influential results in the study of hamiltonicity and by now there are many related known results\,(see, e.g., J. A. Bondy, Basic Graph Theory: Paths and Circuits, Chapter 1 in: {\it Handbook of Combinatorics Vol.1}). A {\it Halin graph} is a planar graph consisting of two edge-disjoint subgraphs: a spanning tree of at least 4 vertices and with no vertex of degree 2, and a cycle induced on the set of the leaves of the spanning tree. Halin graphs possess rich hamiltonicity properties such as being hamiltonian, hamiltonian connected, and almost pancyclic. As a continuous "generalization" of Dirac's theorem, in this paper, we show that there exists a positive integer $n_0$ such that any graph $G$ with $n\ge n_0$ vertices and $\delta(G)\ge (n+1)/2$ contains a spanning Halin subgraph. In particular, it contains a spanning Halin subgraph which is also pancyclic., Comment: 25 pages, 3 figures

Gelfand-Tsetlin polytopes are prominent objects in algebraic combinatorics. The number of integer points of the Gelfand-Tsetlin polytope $\mathrm{GT}(\lambda)$ is equal to the dimension of the corresponding irreducible representation of $GL(n)$. It is well-known that the Gelfand-Tsetlin polytope is a marked order polytope; the authors have recently shown it to be a flow polytope. In this paper, we draw corollaries from this result and establish a general theory connecting marked order polytopes and flow polytopes., Comment: 18 pages, 10 figures

A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$ , if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is obtained from $X^{n}$ by independently flipping each coordinate with probability $\delta$ . This paper is about self-predicting functions, which are those that coincide with their optimal predictor.

A graphic sequence $\pi$ is potentially $H$-graphic if there is some realization of $\pi$ that contains $H$ as a subgraph. The Erd\H{o}s-Jacobson-Lehel problem asks to determine $\sigma(H,n)$, the minimum even integer such that any $n$-term graphic sequence $\pi$ with sum at least $\sigma(H,n)$ is potentially $H$-graphic. The parameter $\sigma(H,n)$ is known as the potential function of $H$, and can be viewed as a degree sequence variant of the classical extremal function ${\rm ex}(n,H)$. Recently, Ferrara, LeSaulnier, Moffatt and Wenger [On the sum necessary to ensure that a degree sequence is potentially $H$-graphic, Combinatorica 36 (2016), 687--702] determined $\sigma(H,n)$ asymptotically for all $H$, which is analogous to the Erd\H{o}s-Stone-Simonovits Theorem that determines ${\rm ex}(n,H)$ asymptotically for nonbipartite $H$. In this paper, we investigate a stability concept for the potential number, inspired by Simonovits' classical result on the stability of the extremal function. We first define a notion of stability for the potential number that is a natural analogue to the stability given by Simonovits. However, under this definition, many families of graphs are not $\sigma$-stable, establishing a stark contrast between the extremal and potential functions. We then give a sufficient condition for a graph $H$ to be stable with respect to the potential function, and characterize the stability of those graphs $H$ that contain an induced subgraph of order $\alpha(H)+1$ with exactly one edge., Comment: 20 pages, 3 figures

International audience; Let $k,d,\lambda \geqslant 1$ be integers with $d\geqslant \lambda $ and let $X$ be a finite set of points in $\mathbb{R}^{d}$. A $(d-\lambda)$-plane $L$ transversal to the convex hulls of all $k$-sets of $X$ is called Kneser transversal. If in addition $L$ contains $(d-\lambda)+1$ points of $X$, then $L$ is called complete Kneser transversal.In this paper, we present various results on the existence of (complete) Kneser transversals for $\lambda =2,3$. In order to do this, we introduce the notions of stability and instability for (complete) Kneser transversals. We first give a stability result for collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ and $\lambda =2,3$. We then present a description of Kneser transversals $L$ of collections of $d+2(k-\lambda)$ points in $\mathbb{R}^d$ with $k-\lambda\geqslant 2$ for $\lambda =2,3$. We show that either $L$ is a complete Kneser transversal or it contains $d-2(\lambda-1)$ points and the remaining $2(k-1)$ points of $X$ are matched in $k-1$ pairs in such a way that $L$ intersects the corresponding closed segments determined by them. The latter leads to new upper and lower bounds (in the case when $\lambda =2$ and $3$) for $m(k,d,\lambda)$ defined as the maximum positive integer $n$ such that every set of $n$ points (not necessarily in general position) in $\mathbb{R}^{d}$ admit a Kneser transversal.Finally, by using oriented matroid machinery, we present some computational results (closely related to the stability and unstability notions). We determine the existence of (complete) Kneser transversals for each of the $246$ different order types of configurations of $7$ points in $\mathbb{R}^3$.

In this paper we determine the chromatic number of graphs with two odd cycle lengths. Let $G$ be a graph and $L(G)$ be the set of all odd cycle lengths of $G$. We prove that: (1) If $L(G)=\{3,3+2l\}$, where $l\geq 2$, then $\chi(G)=\max\{3,\omega(G)\}$; (2) If $L(G)=\{k,k+2l\}$, where $k\geq 5$ and $l\geq 1$, then $\chi(G)=3$. These, together with the case $L(G)=\{3,5\}$ solved in \cite{W}, give a complete solution to the general problem addressed in \cite{W,CS,KRS}. Our results also improve a classical theorem of Gy\'{a}rf\'{a}s which asserts that $\chi(G)\le 2|L(G)|+2$ for any graph $G$., Comment: 26 pages,accepted version for publication in SIAM J. Discrete Math

Gy\'arf\'as, Gy\H{o}ri and Simonovits proved that if a $3$-uniform hypergraph with $n$ vertices has no linear cycles, then its independence number $\alpha \ge \frac{2n} {5}$. The hypergraph consisting of vertex disjoint copies of a complete hypergraph $K_5^3$ on five vertices, shows that equality can hold. They asked whether this bound can be improved if we exclude $K_5^3$ as a subhypergraph and whether such a hypergraph is $2$-colorable. In this paper we answer these questions affirmatively. Namely, we prove that if a $3$-uniform linear-cycle-free hypergraph doesn't contain $K_5^3$ as a subhypergraph, then it is $2$-colorable. This result clearly implies that its independence number $\alpha \ge \lceil \frac{n}{2} \rceil$. We show that this bound is sharp. Gy\'arf\'as, Gy\H{o}ri and Simonovits also proved that a linear-cycle-free $3$-uniform hypergraph contains a vertex of strong degree at most 2. In this context, we show that a linear-cycle-free $3$-uniform hypergraph has a vertex of degree at most $n-2$ when $n \ge 10$., Comment: Improved the writing, more explanation added and corrections made

A permutation rational function $f\in \mathbb{F}_q(x)$ is a rational function that induces a bijection on $\mathbb{F}_q$, that is, for all $y\in\mathbb{F}_q$ there exists exactly one $x\in\mathbb{F}_q$ such that $f(x)=y$. Permutation rational functions are intimately related to exceptional rational functions, and more generally exceptional covers of the projective line, of which they form the first important example. In this paper, we show how to efficiently generate many permutation rational functions over large finite fields using isogenies of elliptic curves, and discuss some cryptographic applications. Our algorithm is based on Fried's modular interpretation of certain dihedral exceptional covers of the projective line (Cont. Math., 1994).

2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that f(0)(P)f(d-1) (P)

Dvo\v{r}\'ak, Kr\'al' and Thomas gave a description of the structure of triangle-free graphs on surfaces with respect to 3-coloring. Their description however contains two substructures (both related to graphs embedded in plane with two precolored cycles) whose coloring properties are not entirely determined. In this paper, we fill these gaps., Comment: 15 pages, 1 figure; corrections from the review process. arXiv admin note: text overlap with arXiv:1509.01013

Lattices are discrete mathematical objects with widespread applications to integer programs as well as modern cryptography. A fundamental problem in both domains is the Closest Vector Problem (popularly known as CVP). It is well-known that CVP can be easily solved in lattices that have an orthogonal basis \emph{if} the orthogonal basis is specified. This motivates the orthogonality decision problem: verify whether a given lattice has an orthogonal basis. Surprisingly, the orthogonality decision problem is not known to be either NP-complete or in P. In this paper, we focus on the orthogonality decision problem for a well-known family of lattices, namely Construction-A lattices. These are lattices of the form $C+q\mathbb{Z}^n$, where $C$ is an error-correcting $q$-ary code, and are studied in communication settings. We provide a complete characterization of lattices obtained from binary and ternary codes using Construction-A that have an orthogonal basis. We use this characterization to give an efficient algorithm to solve the orthogonality decision problem. Our algorithm also finds an orthogonal basis if one exists for this family of lattices. We believe that these results could provide a better understanding of the complexity of the orthogonality decision problem for general lattices.

In 2009, Bang-Jensen asked whether there exists a function $g(k)$ such that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + g(k)$ arcs. In this paper, we answer the question by showing that every strongly $k$-connected $n$-vertex tournament contains a strongly $k$-connected spanning subgraph with at most $kn + 750k^2\log_2(k+1)$ arcs, and there is a polynomial-time algorithm to find the spanning subgraph.

This paper is devoted to a detailed study of nowhere-zero flows on signed eulerian graphs. We generalise the well-known fact about the existence of nowhere-zero $2$-flows in eulerian graphs by proving that every signed eulerian graph that admits an integer nowhere-zero flow has a nowhere-zero $4$-flow. We also characterise signed eulerian graphs with flow number $2$, $3$, and $4$, as well as those that do not have an integer nowhere-zero flow. Finally, we discuss the existence of nowhere-zero $A$-flows on signed eulerian graphs for an arbitrary abelian group~$A$., 17 pages, 1 figure

Benjamini, Shinkar, and Tsur stated the following conjecture on the acquaintance time: asymptotically almost surely ${\mathcal A \mathcal C}(G) \le p^{-1} \log^{O(1)} n$ for a random graph $G \in G(n,p)$, provided that $G$ is connected. Recently, Kinnersley, Mitsche, and Pralat made a major step toward this conjecture by showing that asymptotically almost surely ${\mathcal A \mathcal C}(G) = O(\log n / p)$, provided that $G$ has a Hamiltonian cycle. In this paper, we finish the task by showing that the conjecture holds in the strongest possible sense, that is, it holds right at the time the random graph process creates a connected graph. Moreover, we generalize and investigate the problem for random hypergraphs.

In this paper we show that the pipe dream complex associated to the permutation 1n(n-1)...2 can be geometrically realized as a triangulation of the vertex figure of a root polytope. Leading up to this result we show that the Grothendieck polynomial specializes to the h-polynomial of the corresponding pipe dream complex, which in certain cases equals the h-polynomial of canonical triangulations of root (and flow) polytopes, which in turn equals a specialization of the reduced form of a monomial in the subdivision algebra of root (and flow) polytopes. Thus, we connect Grothendieck polynomials to reduced forms in subdivision algebras and root (and flow) polytopes. We also show that root polytopes can be seen as projections of flow polytopes, explaining that these families of polytopes possess the same subdivision algebra., Comment: 11 pages, 5 figures

Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, mainly due to their generality, it is not clear how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive meta-kernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for $r$-Dominating Set and $r$-Scattered Set on apex-minor-free graphs, and for Planar-\mathcal{F}-Deletion on graphs excluding a fixed (topological) minor in the case where all the graphs in \mathcal{F} are connected., 32 pages

International audience; Phylogenetic networks are used to display the relationship among different species whose evolution is not treelike, which is the case, for instance, in the presence of hybridization events or horizontal gene transfers. Tree inference methods such as maximum parsimony need to be modified in order to be applicable to networks. In this paper, we discuss two different definitions of maximum parsimony on networks, “hardwired” and “softwired,” and examine the complexity of computing them given a network topology and a character. By exploiting a link with the problem Multiterminal Cut, we show that computing the hardwired parsimony score for 2-state characters is polynomial-time solvable, while for characters with more states this problem becomes NP-hard but is still approximable and fixed parameter tractable in the parsimony score. On the other hand we show that, for the softwired definition, obtaining even weak approximation guarantees is already difficult for binary characters and restricted network topologies, and fixed-parameter tractable algorithms in the parsimony score are unlikely. On the positive side we show that computing the softwired parsimony score is fixed-parameter tractable in the level of the network, a natural parameter describing how tangled reticulate activity is in the network. Finally, we show that both the hardwired and the softwired parsimony scores can be computed efficiently using integer linear programming. The software has been made freely available.

Hat problems have recently become a popular topic in combinatorics and discrete mathematics. These have been shown to be strongly related to coding theory, network coding, and auctions. We consider the following version of the hat game, introduced by Winkler and studied by Butler et al. A team is composed of several players; each player is assigned a hat of a given colour; they do not see their own colour, but can see some other hats, according to a directed graph. The team wins if they have a strategy such that, for any possible assignment of colours to their hats, at least one player guesses their own hat colour correctly. In this paper, we discover some new classes of graphs which allow a winning strategy, thus answering some of the open questions in Butler et al. We also derive upper bounds on the maximal number of possible hat colours that allow for a winning strategy for a given graph.

This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite groups, as well as fast Fourier and inverse Fourier transforms for the planar rook monoid, the partial cyclic shift monoid, and the partial rotation monoid., Comment: v2: Streamlined presentation and added results for the planar rook monoid, partial cyclic shift monoid, and partial rotation monoid. 24 pages

A famous conjecture of P\'osa from 1962 asserts that every graph on $n$ vertices and with minimum degree at least $2n/3$ contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Koml\'os, S\'ark\"ozy and Szemer\'edi. In this paper we prove a degree sequence version of P\'osa's conjecture: Given any $\eta >0$, every graph $G$ of sufficiently large order $n$ contains the square of a Hamilton cycle if its degree sequence $d_1\leq \dots \leq d_n$ satisfies $d_i \geq (1/3+\eta)n+i$ for all $i \leq n/3$. The degree sequence condition here is asymptotically best possible. Our approach uses a hybrid of the Regularity-Blow-up method and the Connecting-Absorbing method., Comment: 52 pages, 5 figures, to appear in SIAM J. Discrete Math

We study the effect of boundary conditions on the relaxation time (i.e., inverse spectral gap) of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $\lambda$, called the activity or fugacity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $\omega$ which is the positive solution to $\lambda=\omega(1+\omega)^b$, and vertices are occupied with probability $\omega/(1+\omega)$ when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and nonreconstruction regions at $\omega_r\approx \ln{b}/b$. Reconstruction has been of considerable interest recently since it appears to be intimately connected to the efficiency of local algorithms on locally tree-like graphs, such as sparse random graphs. In this paper we show that the relaxation time of the Glauber dynamics on regular trees $T_h$ of height $h$ with branching factor $b$ and $n$ vertices undergoes a phase transition around the reconstruction threshold. In particular, we construct a boundary condition for which the relaxation time slows down at the reconstruction threshold. More precisely, for any $\omega \le \ln{b}/b$, for $T_h$ with any boundary condition, the relaxation time is $\Omega(n)$ and $O(n^{1+o_b(1)})$. In contrast, above the reconstruction threshold we show that for every $\delta>0$, for $\omega=(1+\delta)\ln{b}/b$, the relaxation time on $T_h$ with any boundary condition is $O(n^{1+\delta + o_b(1)})$, and we construct a boundary condition where the relaxation time is $\Omega(n^{1+\delta/2 - o_b(1)})$. To prove this lower bound in the reconstruction region we introduce a general technique that transforms a reconstruction algorithm into a set with poor conductance.

The classical paradox of social choice theory asserts that there is no fair way to deterministically select a winner in an election among more than two candidates; the only definite collective preferences are between individual pairs of candidates. Combinatorially, one may summarize this information with a graph-theoretic tournament on N vertices (one per candidate), placing an edge from U to V if U would beat V in an election between only those two candidates (no ties are permitted). One well-studied procedure for selecting a winner is to specify a complete binary tree whose leaves are labeled by the candidates, and evaluate it by running pairwise elections between the pairs of leaves, sending the winners to successive rounds of pairwise elections which ultimately terminate with a single winner. This structure is called a voting tree. Much research has investigated which functions on tournaments are computable in this way. Fischer, Procaccia, and Samorodnitsky quantitatively studied the computability of the Copeland rule, which returns a vertex of maximum out-degree in the given tournament. Perhaps surprisingly, the best previously known voting tree could only guarantee a returned out-degree of at least log_2 N, despite the fact that every tournament has a vertex of degree at least (N-1)/2. In this paper, we present three constructions, the first of which substantially improves this guarantee to \Theta(sqrt{N}). The other two demonstrate the richness of the voting tree universe, with a tree that resists manipulation, and a tree which implements arithmetic modulo three., Comment: 13 pages

An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The fractional chromatic number $\chi_f(G)$ is the infimum of $a/b$ over all pairs of positive integers $a,b$ such that $G$ has an $(a:b)$-coloring. Heckman and Thomas conjectured that the fractional chromatic number of every triangle-free graph $G$ of maximum degree at most three is at most 2.8. Hatami and Zhu proved that $\chi_f(G) \leq 3-3/64 \approx 2.953$. Lu and Peng improved the bound to $\chi_f(G) \leq 3-3/43 \approx 2.930$. Recently, Ferguson, Kaiser and Kr\'{a}l' proved that $\chi_f(G) \leq 32/11 \approx 2.909$. In this paper, we prove that $\chi_f(G) \leq 43/15 \approx 2.867$.

In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from `strong stability' forms of the corresponding (pure) extremal results. These results hold for the \alpha-spectral radius defined using the \alpha-norm for any \alpha>1; the usual spectrum is the case \alpha=2. Our results imply that any hypergraph Tur\'{a}n problem which has the stability property and whose extremal construction satisfies some rather mild continuity assumptions admits a corresponding spectral result. A particular example is to determine the maximum \alpha-spectral radius of any 3-uniform hypergraph on n vertices not containing the Fano plane, when n is sufficiently large. Another is to determine the maximum \alpha-spectral radius of any graph on n vertices not containing some fixed colour-critical graph, when n is sufficiently large; this generalizes a theorem of Nikiforov who proved stronger results in the case \alpha=2. We also obtain an \alpha-spectral version of the Erd\H{o}s-Ko-Rado theorem on t-intersecting k-uniform hypergraphs., Comment: 20 pages

We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while the third has a larger dimension., Comment: Changes w.r.t. v4: different style, Section 8 is extended, and in addition a few small changes are made in the rest of the paper. 15 pages, no figures

Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1990. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, permutations, partitions, vector spaces, and designs. One generalization of universal cycles, which require almost complete overlap of consecutive words, is s-overlap cycles, which relax such a constraint. In this paper we study weak orders, which are relations that are transitive and complete. We prove the existence of universal and s-overlap cycles for weak orders, as well as for fixed height and/or weight weak orders, and apply the results to cycles for ordered partitions as well., Comment: 12 pages; final version

The Gyarfas tree packing conjecture states that any set of $n-1$ trees $T_{1},T_{2},\dots, T_{n-1}$ such that $T_i$ has $n-i+1$ vertices packs into $K_n$ (for $n$ large enough). We show that $t=\frac{1}{10}n^{1/4}$ trees $T_1,T_2,\dots, T_t$ such that $T_i$ has $n-i+1$ vertices packs into $K_{n+1}$ (for $n$ large enough). We also prove that any set of $t=\frac{1}{10}n^{1/4}$ trees $T_1,T_2,\dots, T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices packs into $K_{n}$ (for $n$ large enough). Finally, we prove that $t=\frac{1}{4}n^{1/3}$ trees $T_1,T_2,\dots, T_t$ such that $T_i$ has $n-i+1$ vertices packs into $K_n$ as long as each tree has maximum degree at least $2n^{2/3}$ (for $n$ large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Komlos, Sarkozy, and Szemeredi [Combin. Probab. Comput., 10 (2001), pp. 397--416].

In this paper we introduce digital nets over finite abelian groups which contain digital nets over finite fields and certain rings as a special case. We prove a MacWilliams type identity for such digital nets. This identity can be used to compute the strict $t$-value of a digital net over finite abelian groups. If the digital net has $N$ points in the $s$ dimensional unit cube $[0,1]^s$, then the $t$-value can be computed in $\mathcal{O}(N s \log N)$ operations and the weight enumerator polynomial can be computed in $\mathcal{O}(N s (\log N)^2)$ operations, where operations mean arithmetic of integers. By precomputing some values the number of operations of computing the weight enumerator polynomial can be reduced further.

A significant range of geometric structures whose rigidity is explored for both practical and theoretical purposes are formed by modifying generically isostatic triangulated spheres. In the block and hole structures (P, p), some edges are removed to make holes, and other edges are added to create rigid sub-structures called blocks. Previous work noted a combinatorial analogy in which blocks and holes played equivalent roles. In this paper, we connect stresses in such a structure (P, p) to first-order motions in a swapped structure (P', p), where holes become blocks and blocks become holes. When the initial structure is geometrically isostatic, this shows that the swapped structure is also geometrically isostatic, giving the strongest possible correspondence. We use a projective geometric presentation of the statics and the motions, to make the key underlying correspondences transparent., 36 pages, 9 figures

Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curves in surfaces in 3-space. These are balanced rational weighted graphs in tropical surfaces. Specifically, we study the `lifting' problem: given a graph in a tropical surface, can one find a corresponding algebraic curve in a surface? We develop specific combinatorial obstructions to lifting a graph by reducing the problem to the question of whether or not one can factor a polynomial with particular support in the characteristic 0 case. This explains why some unusual tropical curves constructed by Vigeland are not liftable., 19 pages, 2 figures. Revised and reorganized, with a clearer focus on the nature of the combinatorial obstructions

It is an intriguing question to see what kind of information on the structure of an oriented graph $D$ one can obtain if $D$ does not contain a fixed oriented graph $H$ as a subgraph. The related question in the unoriented case has been an active area of research, and is relatively well-understood in the theory of quasi-random graphs and extremal combinatorics. In this paper, we consider the simplest cases of such a general question for oriented graphs, and provide some results on the global behavior of the orientation of $D$. For the case that $H$ is an oriented four-cycle we prove: in every $H$-free oriented graph $D$, there is a pair $A,B\ssq V(D)$ such that $e(A,B)\ge e(D)^{2}/32|D|^{2}$ and $e(B,A)\le e(A,B)/2$. We give a random construction which shows that this bound on $e(A,B)$ is best possible (up to the constant). In addition, we prove a similar result for the case $H$ is an oriented six-cycle, and a more precise result in the case $D$ is dense and $H$ is arbitrary. We also consider the related extremal question in which no condition is put on the oriented graph $D$, and provide an answer that is best possible up to a multiplicative constant. Finally, we raise a number of related questions and conjectures., Comment: 35 pages

A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, matroids, restricted multisets, chains of subsets, multichains, and lattice paths. For subsets, we show that a u-cycle exists for the $k$-subsets of an $n$-set if we let $k$ vary in a non zero length interval. We use this result to construct a "covering" of length $(1+o(1))$$n \choose k$ for all subsets of $[n]$ of size exactly $k$ with a specific formula for the $o(1)$ term. We also show that u-cycles exist for all $n$-length words over some alphabet $\Sigma,$ which contain all characters from $R \subset \Sigma.$ Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets.

We show that if the joints of a bar and joint framework $(G,p)$ are positioned as `generically' as possible subject to given symmetry constraints and $(G,p)$ possesses a `fully-symmetric' infinitesimal flex (i.e., the velocity vectors of the infinitesimal flex remain unaltered under all symmetry operations of $(G,p)$), then $(G,p)$ also possesses a finite flex which preserves the symmetry of $(G,p)$ throughout the path. This and other related results are obtained by symmetrizing techniques described by L. Asimov and B. Roth in their paper `The Rigidity Of Graphs' from 1978 and by using the fact that the rigidity matrix of a symmetric framework can be transformed into a block-diagonalized form by means of group representation theory. The finite flexes that can be detected with these symmetry-based methods can in general not be found with the analogous non-symmetric methods., 26 pages, 10 figures