Sharkovskii proved that, for continuous maps on intervals, the existence of 3-cycle implies the existence of all others. Li and Yorke proved that 3-cycle implies chaos. To establish a domain of uncountable cycles in t...Sharkovskii proved that, for continuous maps on intervals, the existence of 3-cycle implies the existence of all others. Li and Yorke proved that 3-cycle implies chaos. To establish a domain of uncountable cycles in the logistic map and to understand chaos in it, the fixed points of 3-cycle are obtained analytically by solving a sextic equation. At one parametric value, a fixed-point spectrum, resulted from the Sharkovskii limit, helps to realize chaos in the sense of Li and Yorke.展开更多
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 ...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. It is known that if a planar graph G has maximum degree △≥ 9, then ）χ″（G） =△＋ 1. In this paper, we prove that if O is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ″（G） =- 9.展开更多
文摘Sharkovskii proved that, for continuous maps on intervals, the existence of 3-cycle implies the existence of all others. Li and Yorke proved that 3-cycle implies chaos. To establish a domain of uncountable cycles in the logistic map and to understand chaos in it, the fixed points of 3-cycle are obtained analytically by solving a sextic equation. At one parametric value, a fixed-point spectrum, resulted from the Sharkovskii limit, helps to realize chaos in the sense of Li and Yorke.
基金Supported by Natural Science Foundation of Shandong Province(Grant No.ZR2013AM001)
文摘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. It is known that if a planar graph G has maximum degree △≥ 9, then ）χ″（G） =△＋ 1. In this paper, we prove that if O is a planar graph with maximum degree 8 and without a fan of four adjacent 3-cycles, then χ″（G） =- 9.
