Average degrees of edge-chromatic critical graphs

Let G be a simple graph, and let Delta(G), (d) over bar (G) and chi'(G) denote the maximum degree, the average degree and the chromatic index of G, respectively. We called G edge-Delta-critical if chi'(G) = Delta(G)-1- 1 and chi'(H) <= Delta(G) for every proper subgraph H of G. Vizing in 1968 conjectured that if G is an edge -A -critical graph of order n, then (d) over bar (G) >= Delta(G) - 1 + 3/n. We prove that for any edge-Delta-critical graph G, (D) over bar (G) >= min {2 root 2 Delta(G)-30 root 2/2 root 2_1, 3 Delta(G)/4-2}(n), that is, (d) over bar (G) >= {3/4 Delta(G)-22 root 2 Delta(G) -3 -root 2/2 root + 1 approximate to 07388 Delta(G) - 10153 if Delta(G) <= 75; if Delta(G) >= 76. This result improves the best known bound 2/31(Delta(G) + 2) obtained by Woodall in 2007 for Delta(G) >= 41. (C) 2019 Elsevier B.V. All rights reserved.