图 G 的合适的全部的着色以便与 G 的一个周期那些顶点和边事件上有至少 4 种颜色,被称为非循环的全部的着色。G 的非循环的全部的色彩的数字是在 G 的非循环的全部的着色的颜色的最少的数字。在这份报纸,这被证明最大的度的平面图 G 的非循环的全部的色彩的数字至少 k 并且没有 l 周期,是至多(G)+ 2 如果( k , l ){( 6 , 3 ),( 7 , 4 ),( 6 , 5 ),( 7 , 6 )}。
A graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, it is shown that each 1-planar graph with minimum degree 7 contains a copy of K2∨(K1∪K2) with all vertices of degree at most 12. In addition, we also prove the existence of a graph K1∨(K1∪K2 ) with relatively small degree vertices in 1-planar graphs with minimum degree at least 6.