Edge lower bounds for list critical graphs, via discharging
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A graph G is k-critical if G is not (k − 1)-colorable, but every proper subgraph of G is (k − 1)-colorable. A graph G is k-choosable if G has an L-coloring from every list assignment L with |L(v)|=k for all v, and a graph G is k-list-critical if G is not (k−1)-choosable, but every proper subgraph of G is (k−1)-choosable. The problem of determining the minimum number of edges in a k-critical graph with n vertices has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best known lower bound on the number of edges in a k-list-critical graph. In fact, our result on k-list-critical graphs is derived from a lower bound on the number of edges in a graph with Alon–Tarsi number at least k. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area.
Mathematics Subject Classification (2000)05C15
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- R. L. Brooks: On colouring the nodes of a network, Mathematical Proceedings of the Cambridge Philosophical Society, vol. 37, Cambridge Univ Press, 1941, 194–197.Google Scholar
- G. A. Dirac: A theorem of R. L. Brooks and a conjecture of H. Hadwiger, Proceedings of the London Mathematical Society 3 (1957), no. 1, 161–195.Google Scholar
- P. Erdős, A. L. Rubin and H. Taylor: Choosability in graphs, Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing, Congressus Numerantium, vol. 26, 1979, 125–157.Google Scholar
- H. A. Kierstead and L. Rabern: Improved lower bounds on the number of edges in list critical and online list critical graphs, arXiv preprint http://arxiv.org/abs/1406.7355 (2014).Google Scholar
- V. G. Vizing; Vextex coloring with given colors, Metody Diskretnogo Analiza 29 (1976), 3–10 (in Russian).Google Scholar