Blaž Zmazek

Isomorphic components of Kronecker product of bipartite graphs

Weichsel (Proc. Amer. Math. Soc. 13 (1962), 47-52) proved that the Kronecker product of two connected bipartite graphs consists of two connected components. A condition on the factor graphs is presented which ensures that such components are isomorphic. It is demonstrated that several familiar and easily constructible graphs are amenable to that condition. A partial converse is proved for the above condition and it is conjectured that the converse is true in general. Math. Subj. Class. (1991): 05C60 Key terms: Kronecker product, bipartite graphs, graph isomorphism ∗This work was supported in part by the Ministry of Science and Technology of Slovenia under the grants P1-0206-101 and J1-7036.

On a Vizing-like conjecture for direct product graphs

Abstract Let γ ( G ) be the domination number of a graph G , and let G × H be the direct product of graphs G and H . It is shown that for any k ⩾ 0 there exists a graph G such that γ ( G × G ) ⩽ γ ( G ) 2 − k . This in particular disproves a conjecture from [5].

Behzad-Vizing Conjecture and Cartesian Product Graphs.

Abstract We prove the following theorem: if the Behzad-Vizing conjecture is true for graphs G and H, then is it true for the Cartesian product G□H.

Median and quasi-median direct products of graphs

Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G × P3 is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G ×K2 are proved, and it is shown that the only nonbipartite quasimedian direct product is K3 ×K3.