二部平面图的邻点可区别边色数

Adjacent vertex distinguishing index of bipartite planar graphs

图G的邻点可区别边染色是G的一个正常边染色,使得每一对相邻顶点有不同的颜色集合.图G的邻点可区别边色数χ′_a(G)是使得G有邻点可区别边染色的最少颜色数.2006年,Edwards等证明了对最大度至少为12的连通二部平面图,有χ′_a(G)?+1.本文改进了上述结果,证明了若G是最大度至少为7的连通二部平面图,则χ′_a(G)?+1.

A proper edge coloring of a graph G is called adjacent vertex distinguishing,if any pair of adjacent vertices meet distinct color sets.The adjacent vertex distinguishing index of G,denoted by χ′_a（G）,is the smallest integer k such that G has an adjacent vertex distinguishing edge k-coloring.In this paper,we show that every bipartite planar graph G with maximum degree ？ 7 and without isolated edges has χ′_a（G） ？ ＋ 1.This improves a result in Edwards et al.（2006） which asserts that every bipartite planar graph G with ？ 12 and without isolated edges has χ′_a（G） ？ ＋ 1.