A Note on [r,s,c,t]-Colorings of Graphs

Let G be a simple graph with vertex set V(G) and edge set E(G). A subset S of V(G) is called an independent set if no two vertices of S are adjacent in G. The minimum number of independent sets which form a partition of V(G) is called chromatic number of G, denoted by χ(G). A subset S of E(G) is called an edge cover of G if the subgraph induced by S is a spanning subgraph of G. The maximum number of edge covers which form a partition of E(G) is called edge covering chromatic number of G, denoted by χ′c(G). Given nonnegative integers r,s,t and c, an [r,s,c,t]-coloring of G is a mapping f from V(G)⋃E(G) to the color set {0,1,…,k−1} such that the vertices with the same color form an independent set of G, the edges with the same color form an edge cover of G, and |f(vi)−f(vj)|≥r if vi and vj are adjacent, |f(ei)−f(ej)|≥s for every ei,ej from different edge covers, |f(vi)−f(ej)|≥t for all pairs of incident vertices and edges, respectively, and the number of edge covers formed by the coloring of edges is exactly c. The [r,s,c,t]-chromatic number χr,s,c,t(G) of G is defined to be the minimum k such that G admits an [r,s,c,t]-coloring. In this paper, we present the exact value of χr,s,c,t(G) when δ(G)=1 or G is an even cycle.