Edge Clique Covers in Graphs with Independence Number Two: a Special Case

Abstract

The edge clique cover number ecc(G) of a graph G is the size of the smallest set of complete subgraphs whose union covers all edges of G. It has been conjectured that all the simple graphs with independence number two satisfy ecc(G) ≤ n. First, we present a class of graphs containing edges difficult to cover but that satisfy the conjecture. Second, we describe a large class of graphs G such that ecc(G) ≤ 3 2 n. This class is easy to characterize.Definition 3. An edge uv ∈ E(G) is a dominating edge of G if every other vertex of G is adjacent to at least one of u and v.We can now state our first theorem:Theorem 4. Let G be a simple graph with α(G) = 2. If G has a dominating edge uv such that G \ {u, v} has diameter 3, then ecc(G) ≤ n.