The chromatic difference sequence of the Cartesian product of graphs

Abstract

The chromatic difference sequence cds( G) of a graph G with chromatic number n is defined by cds( G) = ( a(1), a(2),…, a( n)) if the sum of a(1), a(2),…, a( t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□ H, has the vertex set V( G□ H = V( G) x V( H) and its edge set is given by ( x 1, y 1)( x 2, y 2) ϵ E( G□ H) if either x 1 = x 2 and y 1 y 2 ϵ E( H) or y 1 = y 2 and x 1 x 2 ϵ E( G). We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.