Independence number of edge-chromatic critical graphs

Let G be a simple graph with maximum degree Delta(G) and chromatic index chi'(G). A classical result of Vizing shows that either chi'(G) = Delta(G) or chi'(G) = Delta(G) + 1. A simple graph G is called edge-Delta-critical if G is connected, chi'(G) = Delta(G) + 1 and chi'(G - e) = Delta(G) for every e is an element of E (G). Let G be an n-vertex edge-Delta-critical graph. Vizing conjectured that alpha G), the independence number of G, is at most n/2. The current best result on this conjecture, shown by Woodall, is alpha(G) < 3n/5 . We show that for any given epsilon is an element of (0, 1), there exist positive constants d(0)(epsilon) and D-0(epsilon) such that if G is an n-vertex edge-Delta-critical graph with minimum degree at least d(0) and maximum degree at least D-0, then alpha(G) < (1/2 + epsilon). In particular, we show that if G is an n-vertex edge-Delta-critical graph with minimum degree at least d and Delta(G) >= (d + 1)(4.5d+11.5), then alpha(G) < {7n/12 if d = 3, 4n/7 if d=4, d + 2 + 3 root(d - 1)d/2d + 4 + 3 root(d - 1)d n < 4n/7 if d >= 19.