[r,s,t]-Colorings Of Graphs

Let G =(V(G), E(G)) be a simple graph, where V(G) and E(G) denote the set of vertices and edges, and △(G) denote the maximum degree of G, respectively.Given non-negative integers r, s and t, an [r, s, t]-coloring of a graph G = (V, E) is a mapping c from V ∪ E to the color set{0,1,..., k - 1} such that|c(vi) - c(vj)|≥ r for every two adjacent vertices υi, υj,|c(ei) - c(ej)|≥s for every two adjacent edges ei, ej, and |c(υi) - c(ej)|≥t for every vertex υi and an incident edge ej, respectively. The minimum k such that G admits an [r, s,t]-coloring is called the [r, s,t]-chromatic number of G and is denoted by Xr,s,t(G). Obviously, [1,0,0]-coloring is a vertex coloring, [0,1,0]-coloring is edge coloring, and [1,1,1]-coloring is total coloring. Since [r, s, t]-coloring is combination and extension of the vertex coloring, edge coloring and total coloring, and the relationship is complex among parameters r, s, t. It is interesting to study the problem of [r, s, t]-colorings. In this paper, we studied [r, s,t]-colorings of wheels and wheel related graphs:(1) We determined exact values and bounds for the [r, s, t]-chromatic number of friendship graphs C3(n).(2) We discussed [r, s, t]-colorings of Jahangir graphs Jn,m when m≥ 3 is odd. We first examined exact and bounds for the [r, s, t]-chromatic number of Jahangir graph Jn,m for min{r, s, t}=0. If min{r, s, t}≥ 1, we gave exact values and bounds for the [r, s,t]-chromatic number of Jn,m with n is even and n is odd, respectively.(3) We studied [r, s, t]-colorings of wheels Wn and multiple wheels NWn. We first proved that if min{r, s,t}=0, and if min{r, s,t}≥1, as well as △(NW2n)> 4, then the [r, s,t]-chromatic number of multiple even wheels NW2n is equal to the [r, s, t]-chromatic number of its spanning friendship graphs 3(Nn).Then we determined exact values and upper bounds of Xr,s,t(W4) for r≥1,s≥1,1≤t≤2s. Finally, we provided exact values and some tight bounds for the [r, s,t]-chromatic number of odd multiple wheels NW2n+1.(4) We determined exact values and bounds for the [r, s, t]-chromatic number of fans Fn and multiple fans NFn. We first gave exact values and bounds of Xr,s,t(NFn)(N≥1) for min{r, s, t}=0. Next, We proved that if min{r, s, t}=0, and if min{r,s,t}≥1, △(NF2n)≥ 6; and △(F2n)=4,s≥t, or s< t≤ 2s and r> s + t, then the [r, s, t]-chromatic number of multiple even fans NF2n is equal to the [r,s,t]-chromatic number of its spanning friendship graphs C3(Nn). Then we gave an upper bound of Xr,s,t(F4) for 1≤s<t≤2s and 1≤ r< s+t. Finally, we determined the exact values and an upper bound for the [r, s, t]-chromatic number of multiple odd fans NF2n+1(N≥1) for min{r, s, t}=1.