完全二部图K_(4,n)(n≥47)的点可区别E-全染色

Vertex-Distinguishing E-Total Coloring of Complete Bipartite Graph with

G是一个简单图,G的一个E-全染色f是指使相邻顶点着不同颜色且每条关联边与它的顶点着以不同颜色的全染色。设f为G的一个E-全染色,对任意x∈V(G),用C(x)表示在f下顶点的颜色以及与x关联的边的颜色所构成的集合。若任意u,v∈V(G),u≠v,有C(u)≠C(v),则称f是图G的点可区别的E-全染色,简称VDET染色。图G的VDET染色所用颜色数目的最小值称为图G的的点可区别E-全色数或简称VDET色数,记为χ_vt^e(G)。讨论并给出了完全二部图K_(4,n)(n≥47)的点可区别E-全色数。

G is a simple graph. A total coloring for G is called an E - total coloring if no two adjacent vertices of G receive the same color, and no edge of G receives the same color as one of its endpoints. For an E -total coloring for a graph G and any vertex of G , let C （x） denotes the set of colors of vertex x and of the edges incident with x, we call C （x） the color set of x. If C（u） ≠ C（v） for any two different vertexs u and v of V（G） , then we say thatfis a vertex - distinguishing E - total coloring of G or a VDET coloring of G for short. The minimum number of colors requires for a VDET coloring of G is denoted by χvt^e and is called the VDET chromatic number of G. The VDET coloring of complete bipartite graph is discussed in this paper and the VDET chromatic number of K4,n （ n ≥ 47 ） is obtained.