Neighbor Sum Distinguishing Edge Colorings Of Simple Graphs

A proper [k]-edge coloring of a graph G is a mapping φ : E(G)â†' C= [k] ={1,2,…，k}，if for each adjacent two edges e1,e2∈E(G), we have φ(e1)≠ φ(e2)，then φ is a proper [k-edge coloring of a graph G. By x’(G), we denote the smallest value k in such a coloring of G. For a proper [A;]-edge coloring φ of a graph G, let Sφ(Ï…) denote the sum of the colors of all incident edges of vertex Ï…, if for (？)uυ∈E(G), we have Sφ(u)≠Sφ(Ï…), then φ is a neighbor sum distinguishing [k]-edge coloring of G. By X’Σ (G), we denote the smallest value k in such a coloring of G. The average degree of a graph G is Συ∈v(G)d(Ï…)/|V(G)|; we denote it by ad(G). The maximum average degree mad(G) of G is the maximum of average degrees of its subgraphs. In this paper, we prove two theorems:Theorem 1 If G is a simple graph without isolated edges and mad(G)< 10/3, then χ’Σ(G)≤k，where k = max{A(G)+3,11}.Theorem 2 (1) If G is a normal planar graph and girth g ≥ 5, then χ’Σ(G)≤k, where k = max{A(G) + 3,10}.(2) If G is a normal planar graph without 4-cycle, then χ’Σ(G)≤k，where k = max{A(G) + 3，13}，when A(G)≠10 and k = max{A(G) + 3,14} = 14, when A(G) = 10.This thesis mainly consists of three chapters.In chapter 1，we introduce some definitions and notations whicn are used in this paper. Second, we mention some relate concepts and results about neighbor distinguishing edge colorings of G. And finally, we list our two theo- rems about neighbor sum distinguishing edge colorings.In chapter 2, we prove theorem 1 by discharging method.In chapter 3, we prove theorem 2 by constructing counter-examples using Euler formula and discharging method.