R-Dynamic Coloring And List R-Dynamic Coloring For Some Graphs

Graph colorings theory has the most important role in discrete mathematics,especially the r-dynamic colorings and list r-dynamic coloring get more and more attention.have been widely studied in various fields.An r-dynamic k-coloring of G is a proper k-coloring such that any vertex v has at least min{r,d(v)} distinct colors in NG(v).The r-dynamic chromatic number ?r(G)of a graph G is the least number k such that G has a r-dynamic coloring with k colors.We use ?d(G)denote dynamic chromatic number of G for r = 2.The list dynamic chromatic number chd(G)of G is the least number k such that for any assignment of k-clement.lists to the vertices of G,there is a dynamic coloring of G where the color on each vertex is chosen from its list.The list r-dynamic chromatic number chrd(G)of G is the least number k such that for any assignment of k-element lists to the vertices of G,there is a r-dynamic coloring of G where the color on each vertex is chosen from its list.This articl mainly consists of four chapters:.In chapter one,we introduce some preliminaries,the necessary symbols and the relative studying background of the dynamic colorings and the list r-dynamic colorings.In chapter two,we know that the list chromatic number of the subgraph of graph G and graph G are not inclusion relations,use the comprehensive method we get the ranges of the list r-dynamic chromatic number of G-e and G-v on the basis of the list r-dynamic chromatic number of G are respectively obtained.In chapter three,firstly,we get the range of r-dynamic chromatic number of the direct product of square of path and path,then the direct product of square of path and path is partially colored to obtain the exact values of r-dynamic chromatic number,at the end of the chapter we get the dynamic chromatic number of the direct product of cycle and cycle and the 3-dynamic chromatic number of the direct product of cycle and 4-cycle by using the graph isomorphism.In the fourth chapter.we summarize the main point of the article.put forward the difficulties and deficiencies.The conclusions of this article are new,correct and efficient.And the conclusions provide the the theoretical foundation for the colorings of more complicated graphs.