On the ultimate normalized chromatic difference sequence of a graph

AbstractFor graphs G and H, the Cartesian product G × H is defined as follows: the vertex set is V(G) × V(H), and two vertices (g,h) and (g′,h′) are adjacent in G × H if either g = g′ and hh′ ϵ E(H) or h = h′ and gg′ ϵ E(G). Let Gk denote the Cartesian product of k copies of G. The chromatic difference sequence cds(G) is defined by cds(G) = (a1, a2 − a1,…, at − at − 1,…) where at denotes the maximum number of vertices of t-colorable subgraph of G. The normalized chromatic difference sequence ncds(G) is defined by ncds(G) = cds(G)/V(G). This paper studies the ultimate normalized chromatic difference sequence of a graph NCDS(G) which is equal to the limit of ncds(Gk) as k goes to infinity. We study NCDS(G) under the context of other graph theoretical properties: star chromatic number, hom-regularity, and graph homomorphism. We have provided new upper and lower bounds for NCDS(G). We have also proved, among others, that if there is a homomorphism from a graph G to a graph H, then NCDS(G) dominates NCDS(H)