Coloring Of Planar Graphs And The Applications Of Chromatic Polynomial In The Potts Model

This paper mainly studies the coloring problem about the planar graph,according to the discussion of the chromatic polynomial of zeros problem,to calculate the number of colors in the n region of the flat ring and the number of colors after the partition,also some relevant conclusions are drawn.And because the figure of the break-up of the chromatic polynomial and the partition function of Potts model function and polynomial of knot has certain connection,this article apply chromatic polynomial in statistical mechanics and knot.The main research method of this paper is to provide a new channel for the coloring of planar graphs,that is to say,to discuss the coloring number of the N region graph and its subdivision graph in the ring by calculating the chromatic polynomial of the graph.And we can sum up some conclusions by the comparison of the number of changes before and after the partition graph.To complete this paper successfully,firstly,review the knowledge of graph theory and knot theory and statistical mechanics,and some special graphs are discussed separately.The first discuss the minimum number of coloring of N region graphs.The second generalize the region graph and discuss the change in the number of split graph coloring and give some corresponding examples.The third is that the two flat rings are combined to make the two regionsG_t andG_s have a common edge and discuss the number of colored combination graphs and generalized subdivision graph.The forth combine the three flat rings and discuss the chromatic number of the three graphs,_G~m,_G~n,_tG,which has a common edge for every two of the three graphs.The fifth is triangulation of the region graph and discuss the change of the number of the subdivision graph coloring.Finally,we build the model on the figure in the region of the ring on the basis of the connection between the chromatic polynomial and Potts model split function.According to the discussion of the number of coloring before and after the partition of the planar graph study the corresponding change of the partition function of Potts model.We can also calculate the corresponding chromatic polynomial of knots by the chromatic polynomial of knots between square brackets and polynomial relation and draw some conclusions.