ABSTRACT:A hypergraph is b-simple if no two distinct edges share more than b vertices. Let m (r, t, g) denote the minimum number of edges in an r-uniform non-t-colorable hypergraph of girth at least g.Erdős and Lovász proved thatA result of Szabó improves the lower bound by a factor of r 2− for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b-simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b, t, and > 0 and sufficiently large r, every r-uniform b-simple hypergraph H with maximum edge-degree at most t r r 1− is t-colorable. Some results hold for list coloring, as well.

Given a set F of graphs, a graph G is F-free if G does not contain any member of F as an induced subgraph. We say that F is a degree-sequence-forcing set if, for each graph G in the class C of F-free graphs, every realization of the degree sequence of G is also in C. We give a complete characterization of the degree-sequence-forcing sets F when F has cardinality at most two.

A colouring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by χ CF (H). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph H with m edges, χ CF (H) is at most of the order of rm 1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

For a hypergraph G and a positive integer s, let χ (G, s) be the minimum value of l such that G is L-colorable from every list L withThis parameter was studied by Kratochvíl, Tuza, and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of χ (G, s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs. C 2013 Wiley Periodicals, Inc. J. Graph Theory 76: [129][130][131][132][133][134][135][136][137] 2014

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All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any ℓ ∈ {3, 4, 5, 6, 7}, a planar graph is 4-choosable if it is ℓ-cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ ∈ {5, 6, 7} that a planar graph is (4, 2)-choosable if it does not contain chorded ℓ-cycles.

A star k-coloring is a proper k-coloring where the union of two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an I,F-partition. We use a combination of potential functions and discharging to prove that every graph with maximum average degree less than 5 2 has an I,F-partition, which is sharp and answers a question of Cranston and West [A guide to the discharging method, arXiv:1306.4434]. This result implies that planar graphs of girth at least 10 are star 4-colorable, improving upon previous results of Bu, Cranston, Montassier, Raspaud, and Wang [Star coloring of sparse graphs, J. Graph Theory 62 (2009), 201-219].

Let k be a positive integer and n = ⌊log 2 k⌋. We prove that there is an ε = ε(k) > 0 such that for sufficiently large r, every r-uniform hypergraph with maximum edge degree at mostis k-colorable.