A simple lower bound on edge coverings by cliques

Assume that G = G(V, E) is an undirected graph with vertex set V and edge set E. A clique of G is a complete subgraph. An edge clique-covering is a family of cliques of G which cover all edges of G. The edge clique-cover number, Be( G), is the minimum number of cliques in an edge clique-cover of G. For results and applications of the edge clique-cover number see [1-4]. Observe that Be(G) does not change if isolated vertices are removed from G. We give another obvious operation on G which does not effect Be(G). Call vertices x, y equivalent if xy E E and for all vertices z different from x and y, zx E E if and only if zy E E. If x and y are equivalent vertices of G and xy is not an isolated edge then Be( G)= Be(G'), where G' denotes the graph we get from G by identifying x andy. Due to these observations it is enough to determine or estimate Be( G) for graphs without isolated or equivalent vertices.