Neighbor Sum Distinguishing Coloring Of Graphs

In the thesis,all graphs considered are finite,undirect and simple.Let G be a graph with vertex set V(G)and edge set E(G).A k-total coloring of a graph G is a map ?:V(G)U E(G)?{1,2,...,k}.A k-total coloring of G is proper,if it satisfies any two adjacent or incident elements of G have different colors.Similarly,we can define proper edge coloring.Let 0 be a proper k-edge coloring of the graph G and f(v)denote the sum of the colors on all incident edges of v,that is f(v)=?e(?)?(e).We call 0 neighborsum distinguishing k-edge coloring of G if f(u)? f(v)for each edge uv ? E(G).In such a coloring,the smallest value k is called neighbor sum distinguishing edge chromatic number,denoted by ?'?(G).Similarly,let ? be a proper k-total coloring of the graph G and g(v)=?(v)+ ?e(?)?(e)if g(u)?g(v)for each edgeuv ? E(G),then we call ? a neighbor sum distinguishing k-total coloring of graph G,and the smallest value k is called neighbor sum distinguishing total chromatic number,denoted by ?"?(G).We say that a map L is an edge assignment for the graph G if it assigns a list Le of possible colors to each edge e ? E(G).If G has a neighbor sum distinguishing edge coloring ? such that ?(e)? Le for every edge e ? E(G),then we say that G is neighbor sum distinguishing L-edge colorable or 0 is a neighbor sum distinguishing L-edge coloring of G.The graph G is neighbor sum distinguishing k-edge choosable if it is neighbor sum distinguishing L-edge colorable for every edge assignment L satisfying |Le| ? k for each edge e ? E(G).The neighbor sum distinguishing edge choosability of G,denoted by ch'?(G),is the smallest integer k such that G is neighbor sum distinguishing k-edge choosable.The neighbor sum distinguishing total choosability ch"?(G)of G can be defined similarly.It follows directly from the definition that ch'?(G)? ?'?(G),ch"?(G)? ?"?(G).The aim of this thesis is to discuss neighbor sum distinguishing edge coloring,neighbor sum distinguishing edge choosability,neighbor sum distinguishing total coloring,and neighbor sum distinguishing total choosability of some graphs.By using the famous Combinatorial Nullstellensatz and discharging method we get the following results.Theorem 1 Let G be a Halin graph with ?(G)? 4,then ?'?(G)? ?(G)+ 2.Moreover,if ?(G)? 6,then ?'?(G)? ?(G)+ 1.Theorem 2 Let G be a normal 2-degenerate graph with ?(G)>4,then ch'?(G)? ?(G)+4.Theorem 3 Let G be a 2-degenerate graph,then ?"?(G)? ?(G)+ 3.More-over,if ?(G)>5 then ?"?(G)? ?(G)+ 2.Theorem 4 Let G be a graph with ?(G)= 3 and mad(G)<44/15,then ch"?(G)<6.Theorem 5 Let G be a graph with ?(G)= 3 and mad(G)<12/5,then ch"?(G)<5.Theorem 6 Let G be a graph with ?(G)= 4 and mad(G)<5/2 then ch"?(G)?6.Theorem 7 Let G be a planar graph with ?(G)?11 and mad(G)<5,then ch"?(G)??(G)+ 3.Theorem 8 Let G be a d-degenerate graph and satisfy one of the following terms:(1)d ? 8 and ?(G)? 2d;(2)d ? 9 and ?(G)? 5/2d-5.Then ch"?(G)? ?(G)+ d+1.