Triangle-free graphs whose independence number equals the degree

In a triangle-free graph, the neighbourhood of every vertex is an independent set. We investigate the class s of triangle-free graphs where the neighbourhoods of vertices are maximum independent sets. Such a graph G must be regular of degree d = alpha(G) and the fractional chromatic number must satisfy chi(f)(G) = vertical bar G vertical bar/alpha(G). We indicate that s is a rich family of graphs by determining the rational numbers c for which there is a graph G is an element of s with chi(f)(G) = c except for a small gap, where we cannot prove the full statement. The statements for c >= 3 are obtained by using, modifying, and re-analysing constructions of Sidorenko, Mycielski, and Bauer, van den Heuvel and Schmeichel, while the case c < 3 is settled by a recent result of Brandt and Thomasse. We will also investigate the relation between other parameters of certain graphs in s like chromatic number and toughness. (C) 2009 Published by Elsevier B.V.