The maximum number of edges in 2K2-free graphs of bounded degree

We call a graph 2K2-free if it is connected and does not contain two independent edges as an induced subgraph. The assumption of connectedness in this definition only serves to eliminate isolated vertices. Wagon [6] proved that x(G) ~ w(G)[w(G) + 1]/2 if G is 2Krfree where x(G) and w(G) denote respectively the chromatic number and maximum clique size of G. Further properties of 2K2-free graphs have been studied in [1, 3, 4 and,5]. 2K2-free graphs also arise in the theory of perfect graphs. For example, split graphs and threshold graphs are 2K2-free (see [2]). On the other hand, the strong perfect graph conjecture is open for the class of 2K2-free graphs. In this paper we solve the following extremal problem posed by Bermond et al. in [7] and also by Nesetril and Erdos: What is the maximum number of edges in a 2K2-free graph with maximum degree D? Our principal result asserts that the extremal graph is unique for all D and can be obtained from .the five-cycle by multiplying its vertices. The extremal problem solved here is a special case of a more general conjecture of Erdos and Nesetfil which can be viewed as a variation on Vizing's Theorem: Two edges are said to be strongly independent if there is no edge incident to both edges. They conjecture that if L1( G) = D, the edge set of G can be partitioned into at most 5D I 4 color classes in such a way that any two