图在三种约束条件下的正常全染色

设f:V(G)∪E(G)→{1,2,…,k}是简单图G的一个正常k-全染色.令C(f,u)={f(e):e∈N_e(u)},Cf,u]=C(f,u)∪{f(u)},C_2f,u]=C(f,u)∪{f(x):x∈N(u)}∪{f(u)}.N(u)表示顶点u的邻集,N_e(u)表示与顶点u的相关联的边的集合.令Cf;x]={C(f,x);Cf,x];C_2f,x]},对任意的xy∈E(G),Gf;x]≠Cf;y]表示C(f,x)≠C(f,y),Cf,x]≠Cf,y],C_2f,x]≠C_3f,y]同时成立.对任意的边xy∈E(G),如果有Cf;x]≠Cf;y]成立,则称f是图G的一个k-(3)-邻点可区别全染色(简记为(3)-AVDTC).图G的(3)-邻点可区别全染色中最小的颜色数叫做G的(3)-邻点可区别全色数,记为x_((3)as)″(G).研究了联图,完全二部图的(3)-邻点可区别全染色,得到了它们的(3)-邻点可区别全色数.

On Proper Total Colorings of Graphs with Three Constraints

Let f:V(G)∪E(G) → {1,2,...,k} be a proper k-total coloring of a simple graph G.Set C(f,u) ={f(e):e ∈ Ne(u)},Cf,u] =C(f,u) ∪ {f(u)},C2f,u] =C(f,u) ∪ {f(x):x ∈ N(u)} ∪ {f(u)}.令 Cf;x] ={C(f,x);Cf,x];C2f,x]},For each edge xy ∈ E(G),Cf;x] ≠ Cf;y] denoted C(f,x) ≠ C(f,y),Cf,x] ≠ Cf,y],C2f,x] ≠ C2 f,y] holding at the same time.We call f to be a k-(3)-adjacent vertex distinguishing total coloring (k(3)-AVDTC for short) of G if Cf;x] ≠ Cf;y] for each edge xy ∈ E(G).The minimum number of k colors required for which G admits a k-(3)-AVDTC is denoted by x"(3)as(G),and called the (3)-AVDTC chromatic number of G.In this paper,we consider (3)-adjacent vertex distinguishing total coloring of join and complete bipartite graphs.Finally,we obtain their (3)-adjacent vertex distinguishing total chromatic number.