Interval Total Colorings Of Some Cartesian Products Of Graphs

A total coloring of a graph G is a coloring of its vertices and edges such that no adjacet vertices,edges,and no incident vertices and edges obtain the same color. An interval total t-coloring of a graph G is a total coloring of G with colors 1,2,...,t such that all colors are used,and the edges incident to each vertex v together with v are colored by dG(?)+1 consecutive colors,where dG(?)is the degree of a vertex ? in G. A graph G is interval total colorable if it has an interval total t-coloring for some positive integer t.For an interval total colorable graph G,the least and the greatest values of t for which G has an interval total t-coloring are denoted by ?T(G) and WT(G), respectively.In this paper interval total colorings of some Cartesian products of graphs are investigated.First we prove that the Cartesian product of two paths is interval total colorable.Then we show that the Cartesian product of an interval total colorable ?-regular graph G with a path Pm(m?2)or an even cycle C2n(n?2)is interval total colorable, too. Furthermore,we provide the bounds or the exact values of ?T(H) and WT(H),where H is Pm×Pn,G×Pm or G×C2n,and G is an interval total colorable ?-regular graph,m,n?2.