| Given a positive integer k,A proper k edge coloring of a graph(7)EVG,(8)is a mapping c ?{.....2,1:kE} such that every pair of adjacent edges receive different numbers.A proper k total coloring of a graph(7)EVG,(8)is a mapping V U?{.....2,1:kEc} such that every pair of adjacent or incident elements receive different numbers.In a proper edge k coloring of G,let let ?(7)vc(8)denote the sum of colors of all incident edges of vertex v,if for ?(1)(7)GEuv(8),(7)(8)?1?(7)vcuc(8),then we call such a edge k coloring neighbor sum distinguishing k edge coloring,the smallest k for which such a coloring of G exists is the neighbor sum distinguishing edge chromatic number,denoted by(7)G(8)?c￠.Similarly in a proper k total coloring of G,let let ?(7)vs(8)denote the sum of colors of all incident edges of vertex v and the color of vertex v,if for ?(1)(7)GEuv(8),(7)(8)?1?(7)vsus(8),then we call such a total k coloring neighbor sum distinguishing k total coloring,the smallest k for which such a coloring of G exists is the neighbor sum distinguishing total chromatic number,denoted by(7)G(8)?c￠.In this paper,we prove two theorems,the first theorem,For a simple graph G without isolated edges,if(7)Gmad(8)(27)310,then ￠(7)(8)kG ￡?c,where k{(10)(35)(28)14,2max}.the second theorem,For a planar graph G,if(7)G(8)3(35)5 and(7)Gg(8)35,then ￠(7)(8)(7)(8)(10)(35)￡3?cGG.This thesis mainly consists of four chapters.In the first chapter,we we give the background of interrelated filed,some basic notions and present situation.In the second chapter,we prove the first theorem by Euler formula and discharging method.In the third chapter,we prove the second theorem by constructing counter-examples using discharging method.In the fourth chapter,we mainly summarize the conclusions briefly and give the prospects. |
