| This paper mainly studies the polynomial and coloring problem of some plane graphs.It provides a new approach to study the coloring of plane graphs,that is to say,by calculating the polynomial of the dual graph of plane graphs,and discussing the zeros of the polynomial,so obtain the coloring number rule of these plane graphs.This paper reviews the related knowledge of knot theory and graph theory,and defines a class of two adjacent-n mix plane graphG[m 1]（10）[m 2]（10）（42）（10）[m n].Starting from n（28）2,to explore the coloring rule ofG[m 1]（10）[m 2].Then the coloring rule of theG[m 1]（10）[m 2]（10）[m 3]is explored.Finally,the coloring rule of theG[m 1]（10）[m 2]（10）（42）（10）[m n]is calculated by the same method.Whenm1,m2,（42）,mn are all even,at least two or more colors are needed to make it colored.When at least one of m1,m2,（42）,mnis odd,at least three or more colors are needed to make it colored.The plane graphs are then subdivided.Calculate and compare the changing rule of the coloring number of the graphs after subdivision.The paper only provides the calculation process of the G[m 1]（10）[m 2].TheG[m 1]（10）[m 2]is generalized subdivision.The minimum coloring number of the generalized graph will increase by 1 whenm1,m2are all even.In other cases,the minimum coloring number of it will remain unchanged.TheG[m 1]（10）[m 2]is triangulated.The minimum coloring number of the triangulated graph will decrease by 1 when at least one ofm1,m2is odd.In other cases,the minimum coloring number of it will remain unchanged.Apply the above conclusion to knot theory.The coloring of most knots in the knot table is explained theoretically.By using the relationship between square bracket polynomial of the knot and the chromatic polynomial of the graph,some representative chromatic polynomials of some knots are calculated.We analyze the coloring of the knots.And the results are compared with those obtained from the above conclusions.It can reflect the necessity and value of this research. |
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