On The R-hued Coloring Of Graphs

This paper mainly introduces and studies the idea of the r-hued coloring chro-matic of graphs, some results in some bibliographies will been extended. This paper studies the problems of 2- hued,3 -hued and 4-hued coloring of graphs, which included the upper bound of 3-hued and 4- hued chromatic number of a certain fam-ilies of graphs, the relationship between dynamic chromatic number and the function, from different views by inductive discussion method, proof by contradiction, minimal counterexample method.In the first chapter, firstly, the background of graph theory and r- hued coloring of graphs have been introduced and studied briefly. Secondly, we study the status and progress of the r- hued coloring of graphs in this paper. Finally, the definitions, terminology, and notion used throughout the section are detailed described.In the second chapter, we will give the upper bound of 4 - hued chromatic number of a certain families of graphs under certain conditions by the induction hypothesis on the vertex and classifies discussing. Meanwhile, the theorem in this section and some existing conclusions have been compared.In the third chapter, by definition a new function, the relationship between the new function and the dynamic coloring of graphs has been built. In the section, the upper bound of 3 - hued coloring of K1,3- free graphs on the condition of Δ(G)≤ 3 has been proved. Meanwhile, the two theorems in this section and some existing conclusions have been compared.The fourth chapter mainly summarizes the conclusions briefly and gives the prospects.