假定 G 是有最大的度的一张平面图。在它被证明那 G 的这份报纸是全部的 --(+ 2 )-choosable 如果(1 ) 7 并且 G 没有邻近的三角形(即，没有二个三角形是有一个普通的边的事件) ；或(2 ) 6 并且 G 没有交叉三角形(即，没有二个三角形是有一个普通顶点的事件) ；或(3 ) 5， G 没有邻近的三角形， G 没为某整数 k 有 k 周期 { 5， 6 } 。
A total k-coloring of a graph G is a coloring of V(G) ∪ E(G) using k colors such that no two adjacent or incident elements receive the same color.The total chromatic number χ〃(G) is the smallest integer k such that G has a total k-coloring.In this paper,it is proved that the total chromatic number of any graph G embedded in a surface Σ of Euler characteristic χ(Σ)≥0 is Δ(G) + 1 if Δ(G)≥10,where Δ(G) denotes the maximum degree of G.
HOU JianFeng 1,2,WU JianLiang 2,LIU GuiZhen 2 & LIU Bin 2 1 Center for Discrete Mathematics,Fuzhou University,Fuzhou 350002,China
A proper vertex coloring of a graph G is linear if the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number lc(G) of the graph G is the smallest number of colors in a linear coloring of G. In this paper, it is proved that every planar graph G with girth g and maximum degree Δ has(1)lc(G) ≤Δ 21 if Δ≥ 9; (2)lc(G) ≤「Δ/2」 + 7 ifg ≥ 5; (3) lc(G) ≤「Δ/2」 + 2 ifg ≥ 7 and Δ≥ 7.
