Dynamic Coloring Of Planar Graphs

All graphs are finite simple graphs in this paper.A proper k-coloring of a graph G is a mapping ?:V(G)?{1,2…,k} such that ?(u)??(v)for every uv?E(G).A(k,r)-hued coloring of a graph G is a proper k-coloring ? such that |?(NG(v))|?min{dG(v),r}for any vertex v.The r-hued chromatic number of G,written ?r(G),is the minimum integer k such that G has a(k,r)-hued coloring.In 2001,Montgomery et al.first introduced the concept of the hued coloring.Later,more and more scholars began to study.By the definition of ?r(G),it follows immediately that ?(G)=?1(G)??2(G)?…??r(G)?…???(G)=??+1(G)=…=?(G2).We can see that the hued coloring is actually a generalization of the proper coloring.In 2011,similar to Wegner's conjecture,Hongjian Lai et al.proposed a conjecture about r-hued coloring of planar graphs:For planar graphs G,if 1?r?3,then ?r(G)?r+3;If 3?r?7,then ?r(G)?r+5;If r?8,then ?r(G)?[3r/2]+1.Focus on this conjecture,this dissertation has studied some problem about the hued coloring of planar graphs.In the first chapter,we introduce some notions and terminologies of graph and sum-marize the research status of the r-hued coloring.In the second chapter,we study the plane graph G without some special cycle,have?r(G)?r+5.In the third chapter,we study if G is a planar graph of girth at least 5 and r? 24,have ?r(G)?r+4.