Search

The question of whether 3-Coloring can be solved in polynomial-time for the diameter two graphs is a well-known open problem in the area of algorithmic graph theory. We study the problem restricted to graph classes that avoid cycles of given lengths as induced subgraphs. Martin et. al. [CIAC 2021] showed that the problem is polynomial-time solvable for $C_5$-free or $C_6$-free graphs, and, $(C_4,C_s)$-free graphs where $s \in \{3,7,8,9\}$. We extend their result proving that it is polynomial-time solvable for $(C_4,C_s)$-free graphs, for any constant $s$, and for $(C_3,C_7)$-free graphs. Our results also hold for the more general problem List 3-Colouring., Comment: 14 pages. Revised version accepted to 18th Algorithms and Data Structures Symposium (WADS 2023)

We show that for any natural number $k \ge 1$, any oriented graph $D$ of minimum semidegree at least $(3k- 2)/4$ contains an antidirected path of length $k$. In fact, a slightly weaker condition on the semidegree sequence of $D$ suffices, and as a consequence, we confirm a weakened antidirected path version of a conjecture of Addario-Berry, Havet, Linhares Sales, Thomass\'e and Reed., Comment: Fixed an accidental omission in Problem 10

Soon after his 1964 seminal paper on edge colouring, Vizing asked the following question: can an optimal edge colouring be reached from any given proper edge colouring through a series of Kempe changes? We answer this question in the affirmative for triangle-free graphs., Comment: 12 pages, 6 figures

The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $H_1,H_2,\ldots$; the graphs in the class are called $(H_1,H_2,\ldots)$-free. The complexity of 3-coloring is far from being understood, even for classes defined by a few small forbidden induced subgraphs. For $H$-free graphs, the complexity is settled for any $H$ on up to seven vertices. There are only two unsolved cases on eight vertices, namely $2P_4$ and $P_8$. For $P_8$-free graphs, some partial results are known, but to the best of our knowledge, $2P_4$-free graphs have not been explored yet. In this paper, we show that the 3-coloring problem is polynomial-time solvable on $(2P_4,C_5)$-free graphs., Comment: 18 pages, 13 figures. Accepted to International Workshop on Graph-Theoretic Concepts in Computer Science (WG) 2021

Let $G(n,n,m)$ be a uniformly random $m$-edge subgraph of the complete bipartite graph $K_{n,n}$ with bipartition $(V_1, V_2)$, where $n_i = |V_i|$. Given a real number $p \in [0,1]$ such that $d_1 := pn_2$ and $d_2 := pn_1$ are integers, let $R(n,n,p)$ be a random subgraph of $K_{n,n}$ such that every $v \in V_i$ has degree $d_i$, for $i = 1, 2$. In this paper we determine sufficient conditions on $n_1,n_2,p$, and $m$ under which one can embed $G(n,n,m)$ into $R(n,n,p)$ and vice versa with probability tending to $1$. In particular, in the balanced case $n_1 = n_2$, we show that if $p \gg \log n/n$ and $1 - p \gg \left(\log n/n \right)^{1/4}$, then for some $m \sim pn^2$, asymptotically almost surely one can embed $G(n,n,m)$ into $R(n,n,p)$, while for $p \gg \left(\log^{3} n/n\right)^{1/4}$ and $1-p \gg \log n/n$ we have the opposite embedding. As an extension, we confirm the Kim--Vu Sandwich Conjecture for degrees growing faster than $(n \log n)^{3/4}$., Comment: Added extension to non-bipartite regular graphs

A graph is called $P_t$-free} if it does not contain a $t$-vertex path as an induced subgraph. While $P_4$-free graphs are exactly cographs, the structure of $P_t$-free graphs for $t \geq 5$ remains little understood. On one hand, classic computational problems such as Maximum Weight Independent Set (MWIS) and $3$-Coloring are not known to be NP-hard on $P_t$-free graphs for any fixed $t$. On the other hand, despite significant effort, polynomial-time algorithms for MWIS in $P_6$-free graphs~[SODA 2019] and $3$-Coloring in $P_7$-free graphs~[Combinatorica 2018] have been found only recently. In both cases, the algorithms rely on deep structural insights into the considered graph classes. One of the main tools in the algorithms for MWIS in $P_5$-free graphs~[SODA 2014] and in $P_6$-free graphs~[SODA 2019] is the so-called Separator Covering Lemma that asserts that every minimal separator in the graph can be covered by the union of neighborhoods of a constant number of vertices. In this note we show that such a statement generalizes to $P_7$-free graphs and is false in $P_8$-free graphs. We also discuss analogues of such a statement for covering potential maximal cliques with unions of neighborhoods.

We construct an infinite family of counterexamples to Thomassen's conjecture that the vertices of every 3-connected, cubic graph on at least 8 vertices can be colored blue and red such that the blue subgraph has maximum degree at most 1 and the red subgraph minimum degree at least 1 and contains no path on 4 vertices.

We show that for every l, there exists d_l such that every 3-edge-connected graph with minimum degree d_l can be edge-partitioned into paths of length l (provided that its number of edges is divisible by l). This improves a result asserting that 24-edge-connectivity and high minimum degree provides such a partition. This is best possible as 3-edge-connectivity cannot be replaced by 2-edge connectivity., Comment: 41 pages, 4 figures

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a prescribed list $L(u) \subseteq \{1,\cdots, k\}$, then we obtain the List $k$-Colouring problem. A graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. We continue an extensive study into the complexity of these two problems for $H$-free graphs. The graph $P_r+P_s$ is the disjoint union of the $r$-vertex path $P_r$ and the $s$-vertex path $P_s$. We prove that List $3$-Colouring is polynomial-time solvable for $(P_2+P_5)$-free graphs and for $(P_3+P_4)$-free graphs. Combining our results with known results yields complete complexity classifications of $3$-Colouring and List $3$-Colouring on $H$-free graphs for all graphs $H$ up to seven vertices., Comment: 20 pages, 6 figures. An extended abstract of this paper appeared in the proceedings of ISAAC 2018

The Barat-Thomassen conjecture, recently proved in [Bensmail et al.: A proof of the Barat-Thomassen conjecture. J. Combin. Theory Ser. B, 124:39-55, 2017.], asserts that for every tree T, there is a constant $c_T$ such that every $c_T$-edge connected graph G with number of edges (size) divisible by the size of T admits an edge partition into copies of T (a T-decomposition). In this paper, we investigate in which case the connectivity requirement can be dropped to a minimum degree condition. For instance, it was shown in [Bensmail et al.: Edge-partitioning a graph into paths: beyond the Barat-Thomassen conjecture. arXiv:1507.08208] that when T is a path with k edges, there is a constant $d_k$ such that every 24-edge connected graph G with size divisible by k and minimum degree $d_k$ has a T-decomposition. We show in this paper that when F is a coprime forest (the sizes of its components being a coprime set of integers), any graph G with sufficiently large minimum degree has an F-decomposition provided that the size of F divides the size of G (no connectivity is required). A natural conjecture asked in [Bensmail et al.: Edge-partitioning a graph into paths: beyond the Barat-Thomassen conjecture. arXiv:1507.08208] asserts that for a fixed tree T, any graph G of size divisible by the size of T with sufficiently high minimum degree has a T-decomposition, provided that G is sufficiently highly connected in terms of the maximal degree of T. The case of maximum degree 2 is answered by paths. We provide a counterexample to this conjecture in the case of maximum degree 3.

Loebl, Koml\'os, and S\'os conjectured that any graph with at least half of its vertices of degree at least k contains every tree with at most k edges. We propose a version of this conjecture for skewed trees, i.e., we consider the class of trees with at most k edges such that the sizes of the colour classes of the trees have a given ratio. We show that our conjecture is asymptotically correct for dense graphs. The proof relies on the regularity method. Our result implies bounds on Ramsey number of several trees of given skew., Comment: 28 pages, 6 figures

In the classic Maximum Weight Independent Set problem we are given a graph $G$ with a nonnegative weight function on vertices, and the goal is to find an independent set in $G$ of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any $P_6$-free graph, that is, a graph that has no path on $6$ vertices as an induced subgraph. This improves the polynomial-time algorithm on $P_5$-free graphs of Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$.

A classical theorem of Erdos, Lovasz and Spencer asserts that the densities of connected subgraphs in large graphs are independent. We prove an analogue of this theorem for permutations and we then apply the methods used in the proof to give an example of a finitely approximable permutation parameter that is not finitely forcible. The latter answers a question posed by two of the authors and Moreira and Sampaio., Comment: 16 pages

Graphons are analytic objects associated with convergent sequences of dense graphs. Finitely forcible graphons, i.e., those determined by finitely many subgraph densities, are of particular interest because of their relation to various problems in extremal combinatorics and theoretical computer science. Lovasz and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon always has finite dimension, which would have implications on the minimum number of parts in its weak eps-regular partition. We disprove the conjecture by constructing a finitely forcible graphon with the space of typical vertices that has infinite dimension.

Fix g>1. Every graph of large enough tree-width contains a g x g grid as a minor; but here we prove that every four-edge-connected graph of large enough tree-width contains a g x g grid as an immersion (and hence contains any fixed graph with maximum degree at most four as an immersion). This result has a number of applications., Comment: 11 pages

We investigate when limits of graphs (graphons) and permutations (permutons) are uniquely determined by finitely many densities of their substructures, i.e., when they are finitely forcible. Every permuton can be associated with a graphon through the notion of permutation graphs. We find permutons that are finitely forcible but the associated graphons are not. We also show that all permutons that can be expressed as a finite combination of monotone permutons and quasirandom permutons are finitely forcible, which is the permuton counterpart of the result of Lovasz and Sos for graphons., Comment: 30 pages, 18 figures

A graph H is strongly immersed in G if G is obtained from H by a sequence of vertex splittings (i.e., lifting some pairs of incident edges and removing the vertex) and edge removals. Equivalently, vertices of H are mapped to distinct vertices of G (branch vertices) and edges of H are mapped to pairwise edge-disjoint paths in G, each of them joining the branch vertices corresponding to the ends of the edge and not containing any other branch vertices. We show that there exists a function d:N->N such that for all graphs H and G, if G contains a strong immersion of the star K_{1,d(Delta(H))|V(H)|} whose branch vertices are Delta(H)-edge-connected to one another, then H is strongly immersed in G. This has a number of structural consequences for graphs avoiding a strong immersion of H. In particular, a class C of simple 4-edge-connected graphs contains all graphs of maximum degree 4 as strong immersions if and only if C has either unbounded maximum degree or unbounded tree-width., Comment: 10 pages

We show that for every hereditary permutation property P and every eps>0, there exists an integer M such that if a permutation p is eps-far from P in the Kendall's tau distance, then a random subpermutation of p of order M has the property P with probability at most eps. This settles an open problem whether hereditary permutation properties are strongly testable, i.e., testable with respect to the Kendall's tau distance. In addition, our method also yields a proof of a conjecture of Hoppen, Kohayakawa, Moreira and Sampaio on the relation of the rectangular distance and the Kendall's tau distance of a permutation from a hereditary property.