完全二部图K_(12,n)(12≤n≤88)的点可区别E-全染色

Vertex-distinguishing E-total coloring of complete bipartite graph K_(12,n) for 12≤n≤88

图G的一个E-全染色是指图G中存在一个映射f:V∪E→{1,2,…,k},对于任意边e=uv∈E(G),有f(e)≠f(u),f(e)≠f(v)且f(u)≠f(v)。在E-全染色f下,令C(v)表示顶点v所染的颜色及与顶点v相邻的边所染的颜色所构成的集合。若∀u,v∈V(G),u≠v有C(u)≠C(v),则称f为图G的k-点可区别E-全染色,简称k-VDET染色。本文证明了完全二部图K12,n分别在12≤n≤28下的6-VDET染色和29≤n≤88下的7-VDET染色。

An E-total coloring of graph G is a mapping f:V∪E→{1,2,…,k}such that for each edge e=uv∈E(G),f(e)≠f(u),f(e)≠f(v)and f(u)≠f(v).For an E-total coloring f of a graph G,let C(v)denote the set of colors of vertex v and the edges incident with v.If C(u)≠C(v)where u,v∈V(G)and u≠v,then that f is a k-vertex-distinguishing E-total coloring of graph G,or simply k-VDET coloring.This paper proves 6-VDET coloring and 7-VDET coloring of complete bipartite graph K12,n for 12≤n≤28 and 29≤n≤88,respectively.