Equality in Vizing’s Conjecture Fixing One Factor of the Cartesian Product

Abstract

In this paper, we investigate the existence of nontrivial solutions for the equation γ(G□H) = γ(G) γ(H) fixing one factor. For the complete bipartite graphs Km,n; we characterize all nontrivial solutions when m = 2, n ≥ 3 and prove the nonexistence of solutions when m, n ≥ 3. In addition, it is proved that the above equation has no nontrivial solution if H is one of the graphs obtained from Cn, the cycle of length n, either by adding a vertex and one pendant edge joining this vertex to any v ∈ V(Cn), or by adding one chord joining two alternating vertices of Cn .