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Coloring, sparseness and girth. (English) Zbl 1344.05058
Summary: An \(r\)-augmented tree is a rooted tree plus \(r\) edges added from each leaf to ancestors. For \(d,g,r \in \mathbb N\), we construct a bipartite \(r\)-augmented complete \(d\)-ary tree having girth at least \(g\). The height of such trees must grow extremely rapidly in terms of the girth.
Using the resulting graphs, we construct sparse non-\(k\)-choosable bipartite graphs, showing that maximum average degree at most \(2(k-1)\) is a sharp sufficient condition for \(k\)-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-\(k\)-colorable graphs and hypergraphs with any girth.

MSC:
05C15 Coloring of graphs and hypergraphs
05C42 Density (toughness, etc.)
05C65 Hypergraphs
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References:
[1] Alon, N., Combinatorial nullstellensatz, combinatorics, Probability and Computing, 8, 7-29, (1999) · Zbl 0920.05026
[2] Alon, N.; Tarsi, M., Colorings and orientations of graphs, Combinatorica, 12, 125-134, (1992) · Zbl 0756.05049
[3] Erdos, P., Graph theory and probability, Canadian Journal of Mathematics, 11, 34-38, (1959) · Zbl 0084.39602
[4] Erdos, P.; Hajnal, A., On chromatic number of graphs and set-systems, Acta Mathematica Academiae Scientiarum Hungaricae, 17, 61-99, (1966) · Zbl 0151.33701
[5] Erdos, P.; Rubin, A. L.; Taylor, H., Choosability in graphs, in Proceedings of the west coast conference onn combinatorics, Graph Theory and Computing (Humboldt State Univ., Arcata, CA., 1979), Congressus Numerantium, 26, 125-157, (1980)
[6] Füredi, Z.; Kostochka, A.; Kumbhat, M., Choosability with separation of complete multipartite graphs and hypergraphs, Journal of Graph Theory, 76, 129-137, (2014) · Zbl 1291.05063
[7] Hakimi, S. L., On the degrees of the vertices of a directed graph, Journal of the Franklin Institute, 279, 290-308, (1965) · Zbl 0173.26404
[8] Kostochka, A. V.; Nesetril, J., Properties of descartes’ construction of trianglefree graphs with high chromatic number, combinatorics, Probability and Computing, 8, 467-472, (1999) · Zbl 0951.05036
[9] Kostochka, A. V.; Stiebitz, M., On the number of edges in colour-critical graphs and hypergraphs, Combinatorica, 20, 521-530, (2000) · Zbl 0996.05046
[10] Kratochvíl, J.; Tuza, Zs.; Voigt, M., Brooks-type theorems for choosability with separation, Journal of Graph Theory, 27, 43-39, (1998) · Zbl 0894.05016
[11] Kriz, I., No article title, A hypergraph free construction of highly chromatic graphs without short cycles, 9, 227-229, (1989)
[12] Lovász, L., On chromatic number of finite set-systems, Acta Mathematica Academiae Scientiarum Hungaricae, 19, 59-67, (1968) · Zbl 0157.55203
[13] Nesetril, J.; Rödl, V., Chromatically optimal rigid graphs, Journal of Combinatorial Theory. B, 46, 133-141, (1989) · Zbl 0677.05031
[14] Richardson, M., Solutions of irreflexive relations, Annals of Mathematics, 58, 573-580, (1953) · Zbl 0053.02902
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