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)∪(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 {pipe}f(v_i)-f(v_j){pipe} ≥ r if v_i and v_j are adjacent, {pipe}f(e_i)-f(e_j){pipe} ≥ s for every e_i, e_j from different edge covers, {pipe}f(v_i) - f(e_j){pipe} ≥ 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 χ_r;s;c;t(G) of G is deffined to be the minimum k such that G admits an [r, s, c, t]-coloring. In this paper, we present the exact value of χ_r,s,c,t(G) when δ(G) = 1 or G is an even cycle.