Coloring On The Graph

The main content of this paper is discussing graph coloring, through discussing the zeros of graphs' chromatic polynomial, it analyzes the number of ways to vertex-color the graph withcorlors so that no two adjacent vertices receive the same color. Graph coloring problem has connection with knot polynomial and the Potts model. Hence, this paper refers to knot theory and statistical mechanics.The purpose of this review is to introduce a new approach about vertex-coloring. We will compute the later graph's dichromatic polynomial when the former graph has removed a part one. Through comparing with the former dichromatic polynomial, some topics will be discussed and we will give some general conclusions. To accomplish this, it is necessary to reformulate and rework the existing and known results in knot theory, graph theory and statistical mechanics. Then illustrative calculations for various families of graphs are presented:The first discussed group of graphs are " Trees around n-circuit"; The second discussed group of graphs are "n-arc-move "; The third discussed group of graphs are "trees-move"; The forth discussed group of graphs are "n-circuit-move"; The fifth discussed group of graphs are "unknown graph+bridge"; The sixth discussed group of graphs are "link-move"; The seventh discussed group of graphs are "unknown graphs around n-circuit".Some implications for correlations of the statistical mechanics are mentioned. The partition function is a useful quantity to determine. It was already proved that various thermodynamic quantities can calculate from the partition function.These cases are also applied to square bracket of the associated alternating knot.