Neighbor Sum Distinguishing Edge Coloring Of Some Graphs

A proper k-edge coloring 0 of a graph G is a mapping ?:E(G)??1,2,…,k}such that each adjacent edges receive different colors.Let f(v)be the sum of colors on the edges incident with v.? is called a k-neighbor sum distinguishing edge coloring if f(u)? f(v)for each edge uv ? E(G).The smallest k such that G has a k-neighbor sum distinguishing edge coloring is called the neighbor sum distinguishing index,denoted by ??'(G).Let ?(G)and mad(G)be the maximum degree and the maximum average degree of a graph G,respectively.In 2011,Flandrin et al.proposed a conjecture for the neighbor sum distinguishing edge coloring,if G is a connected graph on at least 3 vertices and G?C5,then??'(G)??(G)+ 2.In this paper,by using Combinatorial Nullstellensatz and discharging method,we study the neighbor sum distinguishing edge colorings of K4-minor free graphs,a kind of sparse graphs and planar graphs with girth at least 5.We get the following results:(1)Let G be a K4-minor free graph without isolated edges and ?(G)?6.If G has no adjacent vertices of maximum degree,then ??'(G)= ?(G),otherwise,??'(G)= ?(G)+1.(2)Let G be a graph without isolated edges,?(G)? 6 and mad(G)?5/2.If G has no adjacent vertices of maximum degree,then ??'(G)= ?(G),otherwise,??'(G)= ?(G)+ 1.(3)If G is a planar graph with girth at least 5 and without isolated edges,then??'(G)? max{?(G)+ 2,9}.For graphs with maximum degree at least 6 and without isolated edges,if they are K4-minor free graphs or graphs with mad(G)?5/2?Conclusion(1)and Conclusion(2)determine the neighbor sum distinguishing indices;Conclusion(3)implies that the conjecture of neighbor sum distinguishing edge coloring is true for a planar graph G with girth at least 5 and ?(G)? 7.