Luo, Rong; Zhao, Yue An application of Vizing and Vizing-like adjacency lemmas to Vizing’s independence number conjecture of edge chromatic critical graphs. (English) Zbl 1200.05114 Discrete Math. 309, No. 9, 2925-2929 (2009). Summary: In 1968, V.G. Vizing [“Some unsolved problems in graph theory,” Russ. Math. Surv. 23, No.6, 125–141 (1968); translation from Usp. Mat. Nauk 23, No.6(144), 117–134 (1968; Zbl 0177.52301)] conjectured that, if \(G\) is a \(\varDelta \)-critical graph with \(n\) vertices, then \(\alpha (G)\leq \frac n 2\), where \(\alpha (G)\) is the independence number of \(G\). In this note, we apply Vizing and Vizing-like adjacency lemmas to this problem and obtain better bounds for \(\varDelta \in \{7,\dots ,19\}\). Cited in 5 Documents MSC: 05C35 Extremal problems in graph theory 05C15 Coloring of graphs and hypergraphs 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) Keywords:edge coloring; independence number; critical graphs Citations:Zbl 0177.52301 PDFBibTeX XMLCite \textit{R. Luo} and \textit{Y. Zhao}, Discrete Math. 309, No. 9, 2925--2929 (2009; Zbl 1200.05114) Full Text: DOI References: [1] Brinkmann, G.; Choudum, S. A.; Grünewald, S.; Steffen, E., Bounds for the independence number of critical graphs, Bull. London Math. Soc., 32, 137-140 (2000) · Zbl 1020.05050 [2] Grünewald, S.; Steffen, E., Independent sets and 2-factors in edge-chromatic-critical graphs, J. Graph Theory, 45, 113-118 (2004) · Zbl 1033.05041 [3] Jensen, T. R.; Toft, B., Graph Coloring Problems (1995), John Wiley & Sons, Inc. · Zbl 0971.05046 [4] R. Luo, L.Y Miao, Y. Zhao, The size of edge chromatic critical graphs with maximum degree 6, J. Graph Theory (in press); R. Luo, L.Y Miao, Y. Zhao, The size of edge chromatic critical graphs with maximum degree 6, J. Graph Theory (in press) · Zbl 1247.05083 [5] Luo, R.; Zhao, Y., A note on Vizing’s independence number conjecture of edge chromatic critical graphs, Discrete Math., 306, 1788-1790 (2006) · Zbl 1112.05039 [6] Vizing, V. G., Critical graphs with a given chromatic index, Diskretn. Anal., 5, 9-17 (1965), (in Russian) [7] Vizing, V. G., Some unsolved problems in graph theory, Uspekhi Mat.Nauk., 23, 6, 117-134 (1968), (in Russian); or Russian Math. Surveys 23 (6) (1968) 125-141 (in English) · Zbl 0177.52301 [8] Vizing, V. G., The chromatic class of a multigraph, Kibernetika, 3, 29-39 (1965), (in Russian); English tranlation in Cybernetics 1 32-41 · Zbl 0955.05045 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.