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An application of Vizing and Vizing-like adjacency lemmas to Vizing’s independence number conjecture of edge chromatic critical graphs. (English) Zbl 1200.05114

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 \(\varDelta \in \{7,\dots ,19\}\).

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.)

Citations:

Zbl 0177.52301
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References:

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