On Domatic and Total Domatic Numbers of Cartesian Products of Graphs

Abstract

A domatic (resp. total domatic) k-coloring of a graph G is an assignment of k colors to the vertices of G such that each vertex contains vertices of all k colors in its closed neighborhood (resp. open neighborhood). The domatic (resp. total domatic) number of G, denoted d(G) (resp. $d_t (G)$), is the maximum k for which G has a domatic (resp. total domatic) k-coloring. In this paper, we show that for two non-trivial graphs G and H, the domatic and total domatic numbers of their Cartesian product $G \mathbin \square H$ is bounded above by $\max \{|V(G)|, |V(H)|\}$ and below by $\max \{d(G), d(H)\}$. Both these bounds are tight for an infinite family of graphs. Further, we show that if H is bipartite, then $d_t(G \mathbin \square H)$ is bounded below by $2\min \{d_t(G),d_t(H)\}$ and $d(G \mathbin \square H)$ is bounded below by $2\min \{d(G),d_t(H)\}$. As a consequences, we get new bounds for the domatic and total domatic number of hypercubes and Hamming graphs of certain dimensions, and exact values for some n-dimensional tori which turns out to be a generalization of a result due to Gravier from 2002.