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

A simple graph G is called Delta-critical if for chi'(G) - Delta(G) + 1 and chi'(H) <=Delta(G) for every proper subgraph H of G, where Delta(G) and chi'(G) are the maximum degree and the chromatic index of G, respectively. Vizing in 1965 conjectured that any Delta-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 Delta-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 Delta-critical graphs which is called Meredith extension first given by Meredith, we show that if alpha(G') <= (1/2+ f(Delta)) for every Delta-critical graph G' with delta(G') = Delta - 1, thenfor every D-critical graph G with maximum degree D; where f is a nonnegative function of Delta. We also prove that any Delta-critical graph contains a 2-factor if and only if its Meredith extension contains a 2-factor.