Abstract
A simple graph G is called \(\varDelta\)-critical if \(\chi '(G) =\varDelta (G) +1\) and \(\chi '(H) \le \varDelta (G)\) for every proper subgraph H of G, where \(\varDelta (G)\) and \(\chi '(G)\) are the maximum degree and the chromatic index of G, respectively. Vizing in 1965 conjectured that any \(\varDelta\)-critical graph contains a 2-factor, which is commonly referred to as Vizing’s 2-factor conjecture; In 1968, he proposed a weaker conjecture that the independence number of any \(\varDelta\)-critical graph with order n is at most n/2, which is commonly referred to as Vizing’s independence number conjecture. Based on a construction of \(\varDelta\)-critical graphs which is called Meredith extension first given by Meredith, we show that if \(\alpha (G')\le (\frac{1}{2}+f(\varDelta ))|V(G')|\) for every \(\varDelta\)-critical graph \(G'\) with \(\delta (G')=\varDelta -1,\) then \(\alpha (G)<\big (\frac{1}{2}+f(\varDelta )(2\varDelta -5)\big )|V(G)|\) for every \(\varDelta\)-critical graph G with maximum degree \(\varDelta ,\) where f is a nonnegative function of \(\varDelta .\) We also prove that any \(\varDelta\)-critical graph contains a 2-factor if and only if its Meredith extension contains a 2-factor.
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Chen, X., Ji, Q. & Liu, M. Reducing Vizing’s 2-Factor Conjecture to Meredith Extension of Critical Graphs. Graphs and Combinatorics 36, 1585–1591 (2020). https://doi.org/10.1007/s00373-020-02191-8
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DOI: https://doi.org/10.1007/s00373-020-02191-8