On The Several Coloring Problems Of Graph

The coloring problem is one of popular problems in graph theory because of its sig-nifcance in both theory and practice. This paper mainly research on coloring of graphs,More specifcally, we discuss the problems as follows:(1) The coloring problem of3×ngrids which is a hot issue of college entrance examination in recent years;(2) The adjacentvertex distinguishing total coloring, the adjacent strong edge coloring, the equitable ad-jacent strong edge coloring, the adjacent vertex distinguishing vertex edge total coloring,the incidence adjacent vertex distinguishing total coloring and the adjacent vertex stronglydistinguishing total coloring of graph. We obtain corresponding chromatic numbers andgive the specifc coloring schemes. The paper is divided into three chapters:In the frst chapter, we describe the research situation of related felds of graph theoryand the main purpose of the study, and we also show the main results in this paper.In the second chapter, we introduce some defnitions, lemmas, supposes, essentialterms and notations.In the third chapter, we get two new conclusions in the frst segment that each grid isdyed one color and adjacent grids are dyed diferent colors when we use m diferent colorsto dye3×n grids. The new conclusions are obtained by using the latest concept of graphtheory and classifcation analysis of tree structure. In the second segment, we study theadjacent strong edge coloring of double graphs of path, circle and the equitable adjacentstrong edge coloring of double graph of path. Then we obtain corresponding chromatic numbers and give the specifc coloring schemes. In the third segment, we discuss the ad-jacent vertex distinguishing total coloring of necklace, the adjacent vertex distinguishingvertex edge total coloring of necklace and the incidence adjacent vertex distinguishing totalcoloring of necklace. Then we obtain that the corresponding chromatic numbers are sepa-rately fve, three and four. The specifc coloring schemes are also given in this segment. Inthe fourth segment, we research the adjacent vertex strongly distinguishing total coloringof the middle graphs of path, circle and obtain the adjacent vertex strongly distinguishingtotal chromatic numbers. In the ffth segment, we discuss the vertex-distinguishing totalcoloring of Pnk(k=2,3) and the adjacent-vertex-distinguishing total coloring of Cn3.