图的邻点全和可区别全染色

Neighbor full sum distinguishing total coloring of graphs

设f:V(G)∪E(G)→{1,2,…,k}是图G的一个正常k-全染色。令φ(x)=f(x)+eЭx/∑f(e)+∑y∈N(x)/∑f(y),其中N(x)={y∈V(G)|xy∈E(G)}。对任意的边uv∈E(C),若有Φ(u)≠Φ(v)成立,则称f是图G的一个邻点全和可区别k-全染色。图G的邻点全和可区别全染色中最小的颜色数k叫做G的邻点全和可区别全色数,记为f tndi∑(G)。本文确定了路、圈、星、轮、完全二部图、完全图以及树的邻点全和可区别全色数,同时猜想:简单图G(≠K2)的邻点全和可区别全色数不超过△(G)+2。

Let f:V(G)∪E(G)→{1,2,…,k}be a proper k-total coloring of G.Set φ(x)=f(x)+eЭx/∑f(e)+∑y∈N(x)/∑f(y) where N(x)={y∈V(G)|xy∈E(G)}.IfΦ(u)≠Φ(v)for any edge uv∈E(G),then.f is called a k-neighbor full sum distinguishing total coloring of G.The smallest value k for which G has such a coloring is called the neighbor full sum distinguishing total chromatic number of G and denoted by ftndi∑(G).In this paper,we obtain this parameter for paths,cycles,stars,wheels,complete bipartite graphs,complete graphs and trees.Meanwhile,we conjecture that the neighbor full sum distinguishing total chromatic number of C(≠K2)is not more than△(G)+2.