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The transversal number Τ (H) of a hypergraph H is the minimum number of vertices that intersect every edge of H. A 6-uniform hypergraph has all edges of size 6. On 10 November 2000 Tuza and Vestergaard [Discuss. Math. Graph Theory, 22 (2002), pp. 199-210] conjectured that if H is a 3-regular 6-uniform hypergraph of order n, then Τ(H) < 1/4n. In this paper we prove this conjecture, which has become known as the Tuza-Vestergaard conjecture. [ABSTRACT FROM AUTHOR]

A queue layout of a graph consists of a linear order of its vertices and a partition of its edges into queues, so that no two independent edges of the same queue are nested. The queue number of a graph is the minimum number of queues required by any of its queue layouts. A long-standing conjecture by Heath, Leighton and Rosenberg [SIAM J. Discrete Math., 5 (1992), pp. 398--412] states that the queue number of planar graphs is bounded. This conjecture has been partially settled in the positive for several subfamilies of planar graphs (most of which have bounded treewidth). In this paper, we make a further important step towards settling this conjecture. We prove that planar graphs of bounded degree (which may have unbounded treewidth) have bounded queue number. A notable implication of this result is that every planar graph of bounded degree admits a three-dimensional straight-line grid drawing in linear volume. Further implications are that every planar graph of bounded degree has bounded track number, and that every k-planar graph (i.e., every graph that can be drawn in the plane with at most k crossings per edge) of bounded degree has bounded queue number. [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]

A plane graph G is entirely κ-colorable if V (G) ∪ E(G) ∪ F(G) can be colored with κ colors such that any two adjacent or incident elements receive different colors. In 1993, Borodin proved that every plane graph G with maximum degree Δ ≥ 12 is entirely (Δ + 2)-colorable. In this paper, we improve this result by showing that every plane graph G with Δ ≥ 9 is entirely (Δ + 2)-colorable. [ABSTRACT FROM AUTHOR]

This work is a continuation of our previous work [D. M. Ha and H. T. Ngo. Int. J. Math., 34 (2023)], studying the expanding phenomenon in matrices over a finite chain ring with large residue field. A sum-product estimate is proved. It is shown that xy+z+t is a very strong expander, and that x(y + z) and xyz + x are moderate expanders of exponent 13. These results generalize the works of Le and Nguyen (see [J. Number. Theory, 216 (2020), pp. 174--191] and [Discrete Appl. Math., 322 (2022), pp. 166--170]. The solvability in large subsets of the equation xy = z + t is obtained, providing an answer to a question raised by Gyarmati and S\'ark\"ozy [Acta Math. Hungar., 119 (2008), pp. 259--280]. The proofs use spectral graph theory and elementary divisor theory. [ABSTRACT FROM AUTHOR]

An orientation of a graph G is a mod (2p+1)-orientation if, under this orientation, the net out-degree at every vertex is congruent to zero mod 2p+1. If, for any function b : V(G) → ℤ2p+1 satisfying ∑v∈V(G) b(v) ≡ 0 (mod 2p + 1), G always has an orientation D such that the net out-degree at every vertex v is congruent to b(v) mod 2p + 1, then G is strongly ℤ2p+1-connected. The graph G' obtained from G by contracting all nontrivial subgraphs that are strongly ℤ2s+1-connected is called the ℤ2s+1-reduction of G. Motivated by a minimum degree condition of Barat and Thomassen [J. Graph Theory, 52 (2006), pp. 135-146], and by the Ore conditions of Fan and Zhou [SIAM J. Discrete Math., 22 (2008), pp. 288-294] and of Luo et al. [European J. Combin., 29 (2008), pp. 1587-1595] on ℤ3-connected graphs, we prove that for a simple graph G on n vertices, and for any integers s > 0 and real numbers α, β with 0 < α < 1, if for any nonadjacent vertices u,v∈ V(G), dG(u) + dG (v) ≥ αn+ β, then there exists a finite family Ƒ(α, s) of nonstrongly ℤ2s+1-connected graphs such that either G is strongly ℤ2s+1-connected or the ℤ2s+1-reduction of G is in Ƒ(α,s). [ABSTRACT FROM AUTHOR]