Rainbow hamilton cycles in random graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erd?s‐Rényi random graph G_n,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs. In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of G_n,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014     

The distance coloring of graphs

Let G be a connected graph with maximum degree Δ≥ 3.We investigate the upper bound for the chromatic number χγ(G) of the power graph Gγ.It was proved that χγ(G) ≤Δ(Δ-1)γ-1Δ-2+ 1 =:M + 1,where the equality holds if and only if G is a Moore graph.If G is not a Moore graph,and G satisfies one of the following conditions:(1) G is non-regular,(2) the girth g(G) ≤ 2γ- 1,(3)g(G) ≥ 2γ + 2,and the connectivity κ(G) ≥ 3 if γ≥ 3,κ(G) ≥ 4 but g(G) 6 if γ = 2,(4) Δis sufficiently larger than a given number only depending on γ,then χγ(G) ≤ M- 1.By means of the spectral radius λ1(G) of the adjacency matrix of G,it was shown that χ2(G) ≤λ1(G)2+ 1,where the equality holds if and only if G is a star or a Moore graph with diameter 2 and girth 5,and χγ(G)λ1(G)γ+1 ifγ≥3.     

Vertex-distinguishing IE-total Colorings of Cycles and Wheels

Let G be a simple graph. An IE-total coloring f of G refers to a coloring of the vertices and edges of G so that no two adjacent vertices receive the same color. Let C（u） be the set of colors of vertex u and edges incident to u under f. For an IE-total coloring f of G using k colors, if C（u）=C（v） for any two different vertices u and v of V （G）, then f is called a k-vertex-distinguishing IE-total-coloring of G, or a k-VDIET coloring of G for short. The minimum number of colors required for a VDIET coloring of G is denoted by χievt（G）, and is called the VDIET chromatic number of G. We get the VDIET chromatic numbers of cycles and wheels, and propose related conjectures in this paper.     

Sub-exponential graph coloring algorithm for stencil-based Jacobian computations

Partial differential equations can be discretized using a regular Cartesian grid and a stencil-based method to approximate the partial derivatives. The computational effort for determining the associated Jacobian matrix can be reduced. This reduction can be modeled as a (grid) coloring problem. Currently, this problem is solved by using a heuristic approach for general graphs or by developing a formula for every single stencil. We introduce a sub-exponential algorithm using the Lipton–Tarjan separator in a divide-and-conquer approach to compute an optimal coloring. The practical relevance of the algorithm is evaluated when compared with an exponential algorithm and a greedy heuristic.     

Regularity of vertex operator superalgebras

We prove that the regularity of a vertex operator superalgebra can be reduced to the semisimplicity of the category of its weak modules.Moreover,the rationality can be replaced by requiring that each 1/2Z+-graded module is a direct sum of irreducible weak modules.     

给定悬挂点数的双圈图中具有最小Hosoya指数的图

Let G be a graph. The Hosoya index Z（G） of a graph G is defined to be the total number of its matchings. In this paper, we characterize the graph with the smallest Hosoya index of bicyclic graphs with given pendent vertices. Finally, we present a new proof about the smallest Hosoya index of bicyclic graphs.     

若干冠图的邻点可区别V-全染色

应用构造染色函数法研究了冠图C_m·C_n、C_m·C_n的邻点可区别V-全染色.通过对P_m·C_n的邻点可区别V-全染色的研究巧妙给出了C_m·C_n邻点可区别V-全染色,并得到了这些图的邻点可区别V-全色数,从而验证了图的邻点可区别V-全染色猜想.     

具有标准发生率和因病死亡率的离散SIS传染病模型的全局稳定性分析

主要研究了具有标准发生率和因病死亡率的离散SIS传染病模型的动力学性质,利用构造Lyapunov函数,得到模型无病平衡点和地方性平衡点的全局稳定性,即无病平衡点是全局渐近稳定的当且仅当基本再生数R_0≤1,地方病平衡点是全局渐近稳定的当且仅当R_0>1.     

小度数图的邻点可区别全染色

本文研究了最大度为3 且没有相邻最大度的图的邻点可区别全染色. 利用边剖分的方法, 构造了此类图更为一般的情形, 得到了它们的邻点可区别全色数的上界. 目前, 未找到最大度为3 的图且它的邻点可区别全色数是6. 本文的结果部分地回答了这个问题.     

Acyclic edge coloring of triangle-free 1-planar graphs

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by χ a(G), is the least number of colors such that G has an acyclic edge coloring. A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, it is proved that χ a(G) ≤Δ(G) + 22, if G is a triangle-free 1-planar graph.