Neighbor Sum Distinguishing Total Coloring And List Neighbor Sum Distinguishing Total Coloring Of Planar Graphs

Given a graph G ' (V(G), E(G)) and a positive integer k. A total k-coloring of G is a mapping ?: V(G) ? E(G)?{1,2,..., k} such that:(1). ?(u)??(v), for any pair u, v of adjacent vertices;(2). ?(e)??(e'), for any paire, e' of adjacent edges;(3). ?(u)??(e) for any vertex v ? V(G) and any edge e incident with v.Let f(v) denote the sum of the colors of the edges incident with v and the color of v. If for any edge uv ? E(G), f(u) ? f(v), then such a total k-coloring is called neighbor sum distinguishing total coloring. The smallest integer k for which such a coloring of G exists is the neighbor sum distinguishing total chromatic number and denoted by ?tnsd"(G). Let Lz, z ? V ? E be a set of lists of real numbers, such of size k. The neighbor sum distinguishing total choosability of G is the smallest k for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from Lz for each z ? V ? E; and denote it by chtnsd"(G).Such a coloring is called list neighbor sum distinguishing total coloring.For neighbor sum distinguishing total coloring, Pilsniak and Wozniak conjec-tured that: for any simple graph G, ?tnsd"(G)??(G)+3. And the conjecture had been proved holds for complete graphs, cycles, bipartite graphs and subcubic graphs. Clear-ly,?tnsd"(G)?chtnsd"(G),so some papers begin to prove that chtnsd"(G)??(G)+3 also holds for planar graphs.In this paper, two results are proved about neighbor sum distinguishing total col-oring and list neighbor sum distinguishing total coloring of planar graphs by using combinatorial nullstellensatz and discharging method. This paper is divided into four chapters.In Chapter 1, the research background, basic definitions and research status about neighbor sum distinguishing total coloring, list neighbor sum distinguishing total col-oring and other relevant colorings are introduced.In Chapter 2, by studying the structure of planar graphs without 4-cycles with?? 9, it is proved that ?tnsd"(G)??(G)+2 for planar graphs without 4-cycles with?? 9.In Chapter 3, by studying the structure of planar graphs without 5-cycles with chords with ?? 8, it is proved that chtnsd"(G)??(G)+3 for planar graphs without 5-cycles with chords with A > 8.In Chapter 4, conclusions and prospects.