Vertex-Colored Encompassing Graphs

It is shown that every disconnected vertex-colored plane straight line graph with no isolated vertices can be augmented (by adding edges) into a connected plane straight line graph such that the new edges respect the coloring and the degree of every vertex increases by at most two. The upper bound for the increase of vertex degrees is best possible: there are input graphs that require the addition of two new edges incident to a vertex. The exclusion of isolated vertices is necessary: there are input graphs with isolated vertices that cannot be augmented to a connected vertex-colored plane straight line graph.     

Alternating cycles in edge-colored graphs

We show that the edges of a 2-connected graph can be partitioned into two color classes so that every vertex is incident with edges of each color and every alternating cycle passes through a single edge. We also show that the edges of a simple graph with minimum vertex degree δ ? 2 can be partitioned into three color classes so that every vertex is incident with edges in exactly two colors and no cycle is alternating.     

Choosability of planar graphs

A graph G = G(V, E) with lists L(v), associated with its vertices v V, is called L-list colourable if there is a proper vertex colouring of G in which the colour assigned to a vertex v is chosen from L(v). We say G is k-choosable if there is at least one L-list colouring for every possible list assignment L with L(v) = k v V(G).

Now, let an arbitrary vertex v of G be coloured with an arbitrary colour f of L(v). We investigate whether the colouring of v can be continued to an L-list colouring of the whole graph. G is called free k-choosable if such an L-list colouring exists for every list assignment L (L(v) = k v V(G)), every vertex v and every colour f L(v). We prove the equivalence of the well-known conjecture of Erd s et al. (1979): “Every planar graph is 5-choosable” with the following conjecture: “Every planar graph is free 5-choosable”.     

Neighbor Sum Distinguishing Index of Sparse Graphs

A proper k-edge coloring of a graph G is an assignment of one of k colors to each edge of G such that there are no two edges with the same color incident to a common vertex. Let f(v) denote the sum of colors of the edges incident to v. A k-neighbor sum distinguishing edge coloring of G is a proper k-edge coloring of G such that for each edge uv∈E(G), f(u)≠f(v). By χ'_∑(G), we denote the smallest value k in such a coloring of G. Let mad(G) denote the maximum average degree of a graph G. In this paper, we prove that every normal graph with mad(G) ■ and Δ(G) ≥ 8 admits a(Δ(G) + 2)-neighbor sum distinguishing edge coloring. Our approach is based on the Combinatorial Nullstellensatz and discharging method.     

Path covers of weighted graphs

Let (G, w) denote a simple graph G with a weight function w : E(G) ← {0, 1, 2}. A path cover of (G, w) is a collection of paths in G such that every edge e is contained in exactly w(e) paths of the collection. For a vertex v, w(v) is the sum of the weights of the edges incident with v; v is called an odd (even) vertex if w(v) is odd (even). We prove that if every vertex of (G, w) is incident with at most one edge of weight 2, then (G, w) has a path cover P such that each odd vertex occurs exactly once, and each even vertex exactly twice, as an end of a path of P. We also prove that if every vertex of (G, w) is even, then (G, w) has a path cover P such that each vertex occurs exactly twice as an end of a path of P. © 1995 John Wiley & Sons, Inc.     

Acyclic Choosability of Graphs with Bounded Degree

Acta Mathematica Sinica, English Series - An acyclic colouring of a graph G is a proper vertex colouring such that every cycle uses at least three colours. For a list assignment L = {L(v)～ v...     

On heavy paths in 2-connected weighted graphs

A weighted graph is a graph in which every edge is assigned a non-negative real number. In a weighted graph, the weight of a path is the sum of the weights of its edges, and the weighed degree of a vertex is the sum of the weights of the edges incident with it. In this paper we give three weighted degree conditions for the existence of heavy or Hamilton paths with one or two given end-vertices in 2-connected weighted graphs.     

Pointed binary encompassing trees: Simple and optimal

For n disjoint line segments in the plane we construct in optimal O(nlogn) time and linear space an encompassing tree of maximum degree three such that at every vertex v all edges of the tree that are incident to v lie in a halfplane bounded by the line through the input segment which v is an endpoint of. In particular, this tree is pointed since every vertex has an incident angle greater than π. Such a pointed binary tree can be augmented to a minimum pseudo-triangulation. It follows that every set of disjoint line segments in the plane has a constrained minimum pseudo-triangulation whose maximum vertex degree is bounded by a constant.     

关于图的点可区别边染色猜想的一点注

图G的一个k-正常边染色f被称为点可区别的是指任意两点的点及其关联边所染色集合不同,所用最少颜色数被称为G的点可区别边色数,张忠辅教授提出一个猜想即对每一个正整数k≥3,总存在一个最大度为△(G)=k≥3的图G,图G一定有一个子图H,使得G的点可区别的边色数不超过子图的.本文证明了对于最大度△≤6时,猜想正确.     

Linear colorings of subcubic graphs

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induces a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is _C5$C_5$ or _K3,3$K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier and Raspaud [L. Esperet, M. Montassier, and A. Raspaud, Linear choosability of graphs, Discrete Math. 308 (2008) 3938–3950]. Our proof is constructive and yields a linear-time algorithm to find such a coloring.     

具有较小直径的三类图的对极色数

对一个连通图G,令d(u,v)表示G中两个顶点间u和v之间的距离,d表示G的直径.G的一个对极染色指的是从G的顶点集到正整数集(颜色集)的一个映射c,使得对G的任意两个不同的顶点u和v满足d(u,v)+|c(u)-c(v)|≥d.由c映射到G的顶点的最大颜色称为c的值,记作ac(c),而对G的所有对极染色c,ac(c)的最小值称为G的对极色数,记作ac(G).本文确定了轮图、齿轮图以及双星图三类图的对极色数,这些图都具有较小的直径d.     

Optimal double-resource assignment for the robust design problem in multistate computer networks

This paper proposes a novel approach to get the exact optimal double-resource assignment for the robust design problem in multistate computer networks. A multistate computer network consists of links and vertices where both kinds of resources may have several states due to failure, partial failure or maintenance. Therefore, each link (vertex) in the network should be assigned sufficient capacity to keep the network functioning normally. The robust design problem (RDP) in a multistate computer network (MCN) is to search for the minimum capacity assignment of each link and vertex such that the network still survived even under both kinds of failures. However, how to optimally assign the capacity to each resource is not an easy task. This paper proposes an efficient approach to do such assignment and illustrates the efficiency of the proposed approach by some numerical examples.     

Heavy cycles in 2-connected triangle-free weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex v is called the weighted degree of v, denoted by dw(v). The weight of a cycle is defined as the sum of the weights of its edges. Fujisawa proved that if G is a 2-connected triangle-free weighted graph such that the minimum weighted degree of G is at least d, then G contains a cycle of weight at least 2d. In this paper, we proved that if G is a2-connected triangle-free weighted graph of even size such that dw(u) + dw(v) ≥ 2d holds for any pair of nonadjacent vertices u, v ∈ V(G), then G contains a cycle of weight at least 2d.     

Heavy cycles passing through some specified vertices in weighted graphs

A weighted graph is one in which every edge e is assigned a nonnegative number, called the weight of e. The sum of the weights of the edges incident with a vertex υ is called the weighted degree of υ. The weight of a cycle is defined as the sum of the weights of its edges. In this paper, we prove that: (1) if G is a 2‐connected weighted graph such that the minimum weighted degree of G is at least d, then for every given vertices x and y, either G contains a cycle of weight at least 2d passing through both of x and y or every heaviest cycle in G is a hamiltonian cycle, and (2) if G is a 2‐connected weighted graph such that the weighted degree sum of every pair of nonadjacent vertices is at least s, then for every vertex y, G contains either a cycle of weight at least s passing through y or a hamiltonian cycle. AMS classification: 05C45 05C38 05C35. © 2005 Wiley Periodicals, Inc. J Graph Theory     

List Coloring of Random and Pseudo-Random Graphs

choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely whenever . A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least in which no two distinct vertices have more than common neighbors is at most . Received: October 13, 1997     

无向路图和块图上的混合控制

图G=(V,E)的一个混合控制集是一个满足如下条件的集合DV∪E:不在D中的每个点或每条边都相邻或关联于D中的至少一个点或一条边.确定图的最小基数的混合控制集的问题称为混合控制问题.本文研究混合控制问题的算法复杂性,证明了混合控制问题在无向路图上是NP-完全的,但在块图上有线性时间算法.无向路图和块图都是弦图的子类,又是树的母类.     

Fastest mixing Markov chain problem for the union of two cliques

A symmetric, random walk on a graph G can be defined by prescribing weights to the edges in such a way that for each vertex the sum of the weights of the edges incident to the vertex is at most one. The fastest mixing, Markov chain (FMMC) problem for G is to determine the weighting that yields the fastest mixing random walk. We solve the FMMC problem in the case that G is the union of two complete graphs.     

On matching and total domination in graphs

A set M of edges of a graph G is a matching if no two edges in M are incident to the same vertex. A set S of vertices in G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The matching number is the maximum cardinality of a matching of G, while the total domination number of G is the minimum cardinality of a total dominating set of G. In this paper, we investigate the relationships between the matching and total domination number of a graph. We observe that the total domination number of every claw-free graph with minimum degree at least three is bounded above by its matching number, and we show that every k-regular graph with k?3 has total domination number at most its matching number. In general, we show that no minimum degree is sufficient to guarantee that the matching number and total domination number are comparable.     

Coupled choosability of plane graphs

A plane graph G is coupled k‐choosable if, for any list assignment L satisfying for every , there is a coloring that assigns to each vertex and each face a color from its list such that any two adjacent or incident elements receive distinct colors. We prove that every plane graph is coupled 7‐choosable. We further show that maximal plane graphs, ‐minor free graphs, and plane graphs with maximum degree at most three are coupled 6‐choosable. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 27–44, 2008     

Graphoidal bipartite graphs

A graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths inG such that every path in ψ has at least two vertices, every vertex ofG is an internal vertex of at most one path in ψ and every edge of G is in exactly one path in ψ. Let Ω (ψ) denote the intersection graph of ψ. A graph G is said to be graphoidal if there exists a graphH and a graphoidal cover ψof H such that G is isomorphic to Ω(ψ). In this paper we study the properties of graphoidal graphs and obtain a forbidden subgraph characterisation of bipartite graphoidal graphs.