On the graph inequality θE(G)⩾θE(Gm)θE(G)⩾θE(Gm)

Given a simple graph G, the mth power of G  , denoted by GmGm, has the same vertex set as G and an edge joining vertices x and y if and only if there is a walk of length less than or equal to m from x to y  . Let θE(G)θE(G) denote the edge clique cover number of G. In this paper, we present chordal graphs G   satisfying θE(G)<θE(G2)θE(G)<θE(G2), which answers a question proposed by Kang et al. [Graph satisfying inequality θ(G2)⩽θ(G)θ(G2)⩽θ(G), Discrete Math. 250 (2002) 259–264]. We also show that θE(G)⩾θE(G2n+1)θE(G)⩾θE(G2n+1) for any simple graph G and for each positive integer n  , and construct a graph not satisfying θE(G)⩾θE(G2n)θE(G)⩾θE(G2n) for each positive integer n  . Then we give a formula for θE(Tn)θE(Tn) for a tree T   in terms of the number of paths of length n-1n-1 of certain type and the number of pendent vertices of T. Finally we present some open problems.