Graphs satisfying inequality θ(G2)⩽θ(G)

In this paper, we study the edge clique cover number of squares of graphs. More specifically, we study the inequality θ(G2)⩽θ(G) where θ(G) is the edge clique cover number of a graph G. We show that any graph G with at most θ(G) vertices satisfies the inequality. Among the graphs with more than θ(G) vertices, we find some graphs violating the inequality and show that dually chordal graphs and power-chordal graphs satisfy the inequality. Especially, we give an exact formula computing θ(T2) for a tree T.