梯图的邻点可区别均匀Ⅰ-全染色

Incidence-Adjacent Vertex Distinguishing EquitableⅠ-Total Coloring of Ladder Graphs

图的邻点可区别Ⅰ-全染色是指对图的顶点和边染色,使得任意相邻两个顶点的颜色不同,任意相邻两条边的颜色不同,且对任意两个相邻顶点u,v,有C(u)≠C(v),C(u)指该顶点的颜色以及与该点相关联的全体边的颜色构成的集合.图的邻点可区别Ⅰ-全染色如果使得任意两种颜色所染元素数目相差不超过1,则称该染色法为图的邻点可区别均匀Ⅰ-全染色,其所用最少染色数称为图的邻点可区别均匀Ⅰ-全色数.讨论了梯图L_n的邻点可区别均匀Ⅰ-全染色问题,根据该类图的结构性质通过构造有序颜色组,运用循环染色法结合色调整技术,给出它们的邻点可区别均匀Ⅰ-全染色方法,从而有效地确定了其邻点可区别均匀Ⅰ-全色数.

The concept of incidence-adjacent vertex distinguishing equitable total coloring of graph is that when we color vertices and edges of graph, both any two adjacent vertices u and v any two adjacent edges of graph will be colored with different colors, and for any two adjacent vertices u and v, there exists C(u)≠C(v), where C(u) refers to the color set of the vertex u and the set of all colors which are assigned to u and the edges incident to u. An incidence-adjacent vertex distinguishing equitable total coloring of graph is adjacent vertex-distinguishing Ⅰ-total coloring of graph such that the difference of the elements colored by any two colors is not more than 1. The minimum number of colors required in an incidence-adjacent vertex-distinguishing equitable total coloring is called incidence-adjacent vertex distinguishing equitable total chromatic number. The problems of incidence-adjacent vertex distinguishing equitable total coloring of ladder graphs L_n were discussed, and the incidence-adjacent vertex distinguishing equitable total chromatic numbers were confirmed efficiently by using cyclic coloring method combined with permutation method and color adjustment technique based on constructing sequential color group and the structural properties of the graphs.