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The determinant lower bound of Lovász, Spencer, and Vesztergombi [European J. Combin., 7 (1986), pp. 151-160] is a general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovász, Spencer, and Vesztergombi asked if hereditary discrepancy can also be bounded from above by a function of the determinant lower bound. This was answered in the negative by Hoffman, and the largest known multiplicative gap between the two quantities for a set system of m subsets of a universe of size n is on the order of max {log n, √log m}. On the other hand, building upon work of Matousek [Proc. Amer. Math. Soc., 141 (2013), pp. 451-460], Jiang and Reis [in Proceedings of the Symposium on Simplicity in Algorithms (SOSA), SIAM, Philadelphia, 2022, pp. 308-313] showed that this gap is always bounded up to constants by √log(m)log(n). This is tight when m is polynomial in n but leaves open the case of large m. We show that the bound of Jiang and Reis is tight for nearly the entire range of m. Our proof amplifies the discrepancy lower bounds of a set system derived from the discrete Haar basis via Kronecker products. [ABSTRACT FROM AUTHOR]

In this paper, we investigate representations of links that are either centrally symmetric in R3 or antipodally symmetric in S3. By using the notions of antipodally self-dual and antipodally symmetric maps, introduced and studied by the authors in [L. Montejano, J. L. Ramírez Alfonsín, and I. Rasskin, SIAM J. Discrete Math., 36 (2022), pp. 1551-1566], we are able to present sufficient combinatorial conditions for a link L to admit such representations. The latter naturally provide sufficient conditions for L to be amphichiral. We also introduce another (closely related) method yielding again sufficient conditions for L to be amphichiral. We finally prove that a link L, associated to a map G, is amphichiral if the self-dual pairing of G is not one of 6 specific cases among the classification of the 24 self-dual pairing Cor(G)⊳Aut(G). [ABSTRACT FROM AUTHOR]

For two graphs $F$ and $H$, the relative Turán number ${ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau [SIAM J. Discrete Math., 29 (2015), pp. 65--78] and Perarnau and Reed [Combin. Probab. Comput., 26 (2017), pp. 448--467] studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1/\beta = \sum_{i = 2}^r \prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hypergraph $H$ with maximum degree $\Delta$, ${ex}(H,F)\ge \Delta^{-\beta - o(1)} \cdot e(H).$ This is tight as $\Delta \rightarrow \infty$ up to the $o(1)$ term in the exponent, since we show there exists a $\Delta$-regular $r$-graph $H$ such that ${ex}(H,F)=O(\Delta^{-\beta}) \cdot e(H)$. Similar tight results are obtained when $H$ is the random $n$-vertex $r$-graph $H_{n,p}^r$ with edge-probability $p$, extending results of Balogh and Samotij [J. Lond. Math. Soc., 83 (2011), pp. 1091--1094] and Morris and Saxton [Adv. Math., 298 (2016), pp. 534--580]. General lower bounds for a wider class of $F$ are also obtained. [ABSTRACT FROM AUTHOR]

A graph is said to be open-neighborhood conflict-free k-colorable if there exists an assignment of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among the neighbors of v. The open-neighborhood conflict-free chromatic number χO(G) is the smallest k for which G is open-neighborhood conflict-free k-colorable. Z. Abel et al., [SIAM J. Discrete Math. 32 (2018), pp. 2675--2702] showed that χ O(G) ≤ 8 for every planar graph G and posed two open problems to ask whether χO(G) \leq 4 for every planar graph G and whether χO(G) ≤ 3 for every outerplanar graph G. We present in this paper positive answers for the above two open problems by establishing a stronger result which states that, for every integer k ≥ 2, every minor-k-colorable graph is open-neighborhood conflict-free k-colorable, where a graph G is said to be minor-k-colorable if every minor of G is k-colorable. [ABSTRACT FROM AUTHOR]

For a graph G = (V,E), a hypergraph H is called a Berge-G if there is a bijection f : E(G) → E(H) such that e ⊆ f(e) for all e ∈ E(G). The family of Berge-G hypergraphs is denoted by B(G). The maximum number of edges in an n-vertex r-graph with no subhypergraph isomorphic to any Berge-G is denoted by exr(n, B (G)). Gyárfás [SIAM J. Discrete Math., 33 (2019), pp. 383-- 392] showed that for n ≥ 6, ex3(n, B (K4)) = [n/3] [n+1/3] [n+2/3]. However, we found an error in the proof of the result when n ≥ 7. A recent result due to Gerbner, Methuku, and Palmer [European J. Combin., 86 (2020), 103082] implies that for n ≥ 9, ex3(n, B (K4)) = [n/3] [n+1/3] [n+2/3]. In this paper we prove the remaining cases n = 7 and n = 8 for the completeness of the conclusion. [ABSTRACT FROM AUTHOR]

A cut varepsilon-sparsifier of a weighted graph G is a reweighted subgraph of G of (quasi)linear size that preserves the size of all cuts up to a multiplicative factor of varepsilon. Since their introduction by Benczur and Karger [Approximating s-t minimum cuts in O~ (n2) time, in Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing (STOC96), 1996, pp. 47-55], cut sparsifiers have proved extremely influential and found various applications. Going beyond cut sparsifiers, Filtser and Krauthgamer [SIAM J. Discrete Math., 31 (2017), pp. 1263--1276] gave a precise classification of which binary Boolean CSPs are sparsifiable. In this paper, we extend their result to binary CSPs on arbitrary finite domains. [ABSTRACT FROM AUTHOR]

A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak and Tuza [SIAM J. Discrete Math., 7 (1994), pp. 307--313] and Johansson. There have been comparatively fewer works extending these types of bounds to graphs with a small number of triangles. One noteworthy exception is a result of Alon, Krivelevich, and Sudakov [J. Combin. Theory Ser. B, 77 (1999), pp. 73--82] bounding the chromatic number for graphs with low degree and few triangles per vertex; this bound is nearly the same as for triangle-free graphs. This type of parametrization is much less rigid and has appeared in dozens of combinatorial constructions. In this paper, we show a similar type of result for X (G) as a function of the number of vertices n, the number of edges m, as well as the triangle count (both local and global measures). Our results smoothly interpolate between the generic bounds true for all graphs and bounds for triangle-free graphs. Our results are tight for most of these cases; we show how an open problem regarding fractional chromatic number and degeneracy in triangle-free graphs can resolve the small remaining gap in our bounds. [ABSTRACT FROM AUTHOR]

For integers k ≥1 and m ≥2, let g(k,m) be the least integer n ≥1 such that every graph with chromatic number at least n contains a (k + 1)-connected subgraph with chromatic number at least m. Refining the recent result of Gir\~ao and Narayanan [Bull. Lond. Math. Soc., 54 (2022), pp. 868--875] that g(k 1, k) ≤ 7k + 1 for all k≥ 2, we prove that g(k,m) ≤ max(m + 2k 2, (3 + 1/16)k≤) for all k ≥ 1 and m≥2. This sharpens earlier results of Alon et al. [J. Graph Theory, 11 (1987), pp. 367-371], of Chudnovsky [J. Combin. Theory Ser. B, 103 (2013), pp. 567-586], and of Penev, Thomass'e, and Trotignon [SIAM J. Discrete Math., 30 (2016), pp. 592--619]. Our result implies that g(k, k + 1) (3 + 1/16)k for all k \geq 1, making a step closer towards a conjecture of Thomassen [J. Graph Theory, 7 (1983), pp. 261--271] that g(k, k +1)≤3k +1, which was originally a result with a false proof and was the starting point of this research area. [ABSTRACT FROM AUTHOR]

M. A. Henning and A. Yeo conjectured in [SIAM J. Discrete Math., 26 (2012), pp. 687{694] that a bipartite digraph of minimum out-degree at least 3 contains two vertex disjoint directed cycles of different lengths. In this paper, we disprove this conjecture. Further, we classify strong tournaments and strong bipartite tournaments of minimum out-degree 3 without two vertex disjoint directed cycles of different lengths. [ABSTRACT FROM AUTHOR]

We show that for pairs (Q, R) and (S, T) of disjoint subsets of vertices of a graph G, if G is sufficiently large, then there exists a vertex v in V (G) - (Q \cup R \cup S \cup T) such that there are two ways to reduce G by a vertex-minor operation that removes v while preserving the connectivity between Q and R and the connectivity between S and T. Our theorem implies an analogous theorem of Chen and Whittle (SIAM J. Discrete Math., 28 (2014), pp. 1402--1404) for matroids restricted to binary matroids. [ABSTRACT FROM AUTHOR]

This note is an erratum for the paper On the Inequalities of Projected Volumes and the Constructible Region [SIAM J. Discrete Math., 33 (2019), pp. 694--711]. There is a mistake in the proof of Theorem 4 in Section 3. However, Theorem 4 is itself correct (a proof can be found in [Inequalities on Projected Volumes, arxiv.org/abs/1909.12858, 2019] by Leader, Randelovic, and Raty), and all the other results of the paper stand. [ABSTRACT FROM AUTHOR]

We show that the expected size of the maximum agreement subtree of two n-leaf trees, uniformly random among all trees with the shape, is\Theta (\surd n). To derive the lower bound, we prove a global structural result on a decomposition of rooted binary trees into subgroups of leaves called blobs. To obtain the upper bound, we generalize a first moment argument from [D. I. Bernstein, et al., SIAM J. Discrete Math., 29 (2015), pp. 2065--2074] for random tree distributions that are exchangeable and not necessarily sampling consistent. [ABSTRACT FROM AUTHOR]

We revisit the classic problem of estimating the moments of the degree distribution of an undirected simple graph. Consider an undirected simple graph G = (V,E) with n (nonisolated) vertices, and define (for s > 0) Ms =\sum v\in V ds v. Our aim is to estimate Ms within a multiplicative error of (1 +\varepsilon) (for a given approximation parameter\varepsilon > 0) in sublinear time. We consider the sparse-graph model that allows access to uniform random vertices, queries for the degree of any vertex, and queries for a neighbor of any vertex. For the case of s = 1 (the average degree), O\ast (\surd n) queries suffice for any constant\varepsilon [U. Feige, SIAM J. Comput., 35 (2006), pp. 964--984], [O. Goldreich and D. Ron, Random Structures Algorithms, 32 (2008), pp. 473--493]. (We use the O\ast notation to suppress dependencies in log n and 1/\varepsilon.) Gonen, Ron, and Shavitt [SIAM J. Discrete Math., 25 (2011), pp. 1365--1411] extended this result to all integral s > 0 by designing an algorithm that performs O\ast (n1 1/(s+1)) queries. (Strictly speaking, their algorithm approximates the number of star-subgraphs of a given size, but a slight modification gives an algorithm for moments.) We design a new, significantly simpler algorithm for this problem. In the worst case, it exactly matches the bounds of Gonen, Ron, and Shavitt and has a much simpler proof. More importantly, the running time of this algorithm is connected to the arboricity of G. This is (essentially) the maximum density of an induced subgraph. For the family of graphs with arboricity at most\alpha, it has a query complexity of O\ast\bigl(n\cdot\alpha 1/s M 1/s s + min\bigl\{ m M 1/s s, m\cdot ns 1 Ms\bigr\}\bigr) which is always upper bounded by O\ast\bigl(n\alpha M 1/s s\bigr). Thus, for the class of constant-arboricity graphs (which includes, among others, all minor-closed families and preferential attachment graphs), we can estimate the average degree in O\ast (1) queries, and we can estimate the variance of the degree distribution in O\ast (\surd n) queries. This is a major improvement over the previous worst-case bounds. [ABSTRACT FROM AUTHOR]

Carvalho, Lucchesi, and Murty [J. Combin. Theory Ser. B, 92 (2004), pp. 319--324, Theorem 3.5] presented a proof of a theorem of Reed and Wakabayashi that a brick G is nonsolid if and only if there exist two vertex-disjoint odd cycles C1 and C2 such that G V (C1 ∪ C2) has a perfect matching. Consequently, every brick with no two vertex-disjoint odd cycles is solid. Recently, Lucchesi et al. [SIAM J. Discrete Math., 32 (2018), pp. 1478--1501] constructed infinite families of solid bricks containing two vertex-disjoint odd cycles. Noticing that none of these graphs is cubic, they conjectured that no cubic solid brick contains two vertex-disjoint odd cycles. In this note, we present an infinite family of graphs showing that this conjecture fails. We further show that the minimum counterexample is unique, which has 12 vertices. [ABSTRACT FROM AUTHOR]