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.