Hued Colorings Of The Product Graphs

The graph vertex coloring is an important research direction in graph theory,and it has extremely wide application in real life.Such as student electives,four coloring problem,conference scheduling,relay station allocation,especially computer networks,can be translated into vertex coloring problems.A r-hued k-coloring of a graph G is a proper coloring c such that |c(N(v))| ?min{r,d(v)} for every vertex v ? V(G).It includes proper coloring(1-hued coloring),dynamic coloring(2-hued coloring),and squared coloring(?-hued coloring),and is a very interesting new coloring.The minimum chromatic number to determine whether a graph can be r-hued coloring,that is,r-hued chromatic number,is the primary task of the study.Circle and path are the most basic types of graphs,and their basic conclusions are given in[1].The l-power of the graph G is the basic concept of the graph,its vertex is the same as the graph G.and the two vertices are adjacent,if and only if the distance between two vertices in the graph G does not exceed l.We first discuss the hued chromatic number of Pml and Cnl,and study the r-hued chromatic number of Pn2,Cm2.Cartesian product and Strong product are the basic forms of graph operations.In[8],[10],[12,13]the author studied the hued chromatic number of the product graph.On the basis of this,this paper studies r-hued chromatic number of the Cartesian Product graph of Pm2 and Pn,Cm2 and Pn,r-hued chromatic numbers of the Strong Product graph Pn and Pm,Pn and Cm and obtains some good conclusions.In this paper,we obtain the exact.value by proving that the upper and lower bounds of the hued chromatic number and r-hued chromatic number are equal.The upper bound is obtained by applying the relevant algebraic operation to give the specific coloring,and the lower bound is obtained through the counter-evidence and strict reasoning.