Reducing Vizing’s 2-Factor Conjecture to Meredith Extension of Critical Graphs

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.