Compatible connectedness in graphs and topological spaces

A topology on the vertex set of a graphG iscompatible with the graph if every induced subgraph ofG is connected if and only if its vertex set is topologically connected. In the case of locally finite graphs with a finite number of components, it was shown in [11] that a compatible topology exists if and only if the graph is a comparability graph and that all such topologies are Alexandroff. The main results of Section 1 extend these results to a much wider class of graphs. In Section 2, we obtain sufficient conditions on a graph under which all the compatible topologies are Alexandroff and in the case of bipartite graphs we show that this condition is also necessary.     

On geodesic structures of weakly median graphs—II: Compactness, the role of isometric rays

We prove that the vertex set of a K_ℵ0-free weakly median graph G endowed with the weak topology associated with the geodesic convexity on V(G) is compact if and only if G has one of the following equivalent properties: (1) G contains no isometric rays; (2) any chain of interval of G ordered by inclusion is finite; (3) every self-contraction of G fixes a non-empty finite regular weakly median subgraph of G. We study the self-contractions of K_ℵ0-free weakly median graphs which fix no finite set of vertices. We also follow a suggestion of Imrich and Klavzar [Product Graphs, Wiley, New York, 2000] by defining different centers of such a graph G, each of them giving rise to a non-empty finite regular weakly median subgraph of G which is fixed by all automorphisms of G.     

点可迁图的限制边连通性

3限制边割是连通图的一个边割, 它将此图分离成阶不小于3的连通分支. 图G的最小3限制边割所含的边数称为此图的3限制边连通度, 记作λ\-3(G). 它以图G的3阶连通点导出 子图的余边界的最小基数ξ_3(G)为上界. 如果λ_3(G)=ξ_3(G), 则称图G是极大3限制边连通的 . 已知在某种程度上,3限制边连通度较大的网络有较好的可靠性. 作者在文中证明: 如果k正则连通点可迁图的 围长至少是5, 那么它是是极大3限制边连通的.     

Amply regular graphs and block designs

We study the amply regular diameter d graphs Γ such that for some vertex a the set of vertices at distance d from a is the set of points of a 2-design whose set of blocks consists of the intersections of the neighborhoods of points with the set of vertices at distance d-1 from a. We prove that the subgraph induced by the set of points is a clique, a coclique, or a strongly regular diameter 2 graph. For diameter 3 graphs we establish that this construction is a 2-design for each vertex a if and only if the graph is distance-regular and for each vertex a the subgraph Γ₃(a) is a clique, a coclique, or a strongly regular graph. We obtain the list of admissible parameters for designs and diameter 3 graphs under the assumption that the subgraph induced by the set of points is a Seidel graph. We show that some of the parameters found cannot correspond to distance-regular graphs.     

Induced subgraphs with large degrees at end-vertices for hamiltonicity of claw-free graphs

A graph is called claw-free if it contains no induced subgraph isomorphic to K1,3. Matthews and Sumner proved that a 2-connected claw-free graph G is Hamiltonian if every vertex of it has degree at least (|V(G)|-2)/3. At the workshop C&C (Novy Smokovec, 1993), Broersma conjectured the degree condition of this result can be restricted only to end-vertices of induced copies of N (the graph obtained from a triangle by adding three disjoint pendant edges). Fujisawa and Yamashita showed that the degree condition of Matthews and Sumner can be restricted only to end-vertices of induced copies of Z₁ (the graph obtained from a triangle by adding one pendant edge). Our main result in this paper is a characterization of all graphs H such that a 2-connected claw-free graph G is Hamiltonian if each end-vertex of every induced copy of H in G has degree at least |V(G)|/3+1. This gives an affirmative solution of the conjecture of Broersma up to an additive constant.     

Cores of Imprimitive Symmetric Graphs of Order a Product of Two Distinct Primes

A retract of a graph Γ is an induced subgraph Ψ of Γ such that there exists a homomorphism from Γ to Ψ whose restriction to Ψ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph Γ is G‐symmetric if G is a subgroup of the automorphism group of Γ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of Γ admits a nontrivial partition that is preserved by G, then Γ is an imprimitive G‐symmetric graph. In this paper cores of imprimitive symmetric graphs Γ of order a product of two distinct primes are studied. In many cases the core of Γ is determined completely. In other cases it is proved that either Γ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.     

有约束条件的图的(g,f)-因子

设G是一个图,用V(G)和E(G)表示它的顶点集和边集,并设g和f是定义在V(G)上的两个整数值函数且g＜f.图G的一个(g,f)-因子是G的一个支撑子图F使对任意的x∈V(G)有g(x)≤dF(x)≤f(x).如果过图G的任意k条边都有一个(g,f)-因子,则称图G是一个(g,f)-k-覆盖图.如果图G的任意k条边不属于它的一个(g,f)-因子,则称图G是一个(g,f)-k-消去图.作者分别给出了一个图是(g,f)-k-覆盖图和(g,f)-k-消去图的充分条件.     

关于图中子图的（n,k）—正交因子分解

设G是一个具有顶点集V(G)和边集E(G)的图. 设g和f是定义在V(G)上的两个整数值函数,使得g(x)f(x)对所有的点x∈V(G)都成立.如果G是一个(mg+n,mf-n)-图,1n<m2k,且g(x)2k-1对所有的点x∈V(G)都成立,则对任意给定具有|E(H)|=nk边的G的子图H,存在G的一个子图G′使G′有一个(g,f)-因子分解(n,k)-正交H.     

Sources and Sinks in Comparability Graphs

We prove that a subset S of vertices of a comparability graph G is a source set if and only if each vertex of S is a source and there is no odd induced path in G between two vertices of S. We also characterize pairs of subsets corresponding to sources and sinks, respectively. Finally, an application to interval graphs is obtained.     

Interval coloring of (3,4)‐biregular bipartite graphs having large cubic subgraphs

An interval coloring of a graph is a proper edge coloring such that the set of used colors at every vertex is an interval of integers. Generally, it is an NP‐hard problem to decide whether a graph has an interval coloring or not. A bipartite graph G = (A,B;E) is (α, β)‐biregular if each vertex in A has degree α and each vertex in B has degree β. In this paper we prove that if the (3,4)‐biregular graph G has a cubic subgraph covering the set B then G has an interval coloring. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 122–128, 2004     

一类泛圈图

本文证明了如果 Ｇ 是 ２ 连通无爪图, Ｇ 不是圈,ｎ＝ ｜ Ｖ（ Ｇ）｜≥９, Ｇ 的每个导出子图 Ａ都满足φ（ａ１,ａ２ ）,且 Ｇ 中不含同构于 Ｚ＋２ 的导出子图,则 Ｇ是泛圈图     

3阶幂图的广义导出匹配可扩性

A graph G is induced matching extendable if every induced matching of G is included in a perfect matching of G. A graph G is generalized induced matching extendable if every induced matching of G is included in a maximum matching of G. A graph G is claw-free, if G dose not contain any induced subgraph isomorphic to K1,3. The k-th power of G, denoted by Gu, is the graph with vertex set V（G） in which two vertices are adjacent if and only if the distance between them is at most k in G. In this paper we show that, if the maximum matchings of G and G3 have the same cardinality, then G3 is generalized induced matching extendable. We also show that this result is best possible. As a result, we show that if G is a connected claw-flee graph, then G3 is generalized induced matching extendable.     

Independent domination in chordal graphs

A graph chordal if it does not contain any cycle of length greater than three as an induced subgraph. A set of S of vertices of a graph G = (V,E) is independent if not two vertices in S are adjacent, and is dominating if every vertex in V?S is adjacent to some vertex in S. We present a linear algorithm to locate a minimum weight independent dominating set in a chordal graph with 0–1 vertex weights.     

图的（g，f）－因子和因子分解

设Ｇ是一个图，ｇ，ｆ是定义在图Ｇ的顶点集上的两个整数值函数且图Ｇ的一个（ｇ，ｆ）－因子是Ｇ的一个支撑子图Ｆ使对任意的ｘ∈Ｖ（Ｆ）有本文给出了一个图（ｇ，ｆ）－可因子化的若干充分条件和一个图是（ｇ，ｆ）－消去图的充分必要条件，并研究了这些条件的应用。     

图的（g，f）－因子和因子分解

设Ｇ是一个图，ｇ，ｆ是定义在图Ｇ的顶点集上的两个整数值函数且图Ｇ的一个（ｇ，ｆ）－因子是Ｇ的一个支撑子图Ｆ使对任意的ｘ∈Ｖ（Ｆ）有本文给出了一个图（ｇ，ｆ）－可因子化的若干充分条件和一个图是（ｇ，ｆ）－消去图的充分必要条件，并研究了这些条件的应用。     

Regular pseudo-median graphs

A graph is pseudo-median if for every triple u, v, w of vertices there exists either a unique vertex between each pair of them (if their mutual distances sum up to an even number) or a unique triangle whose edges lie between the three pairs of u, v, w, respectively (if the distance sum is odd). We show that a finite pseudo-median graph is regular if and only if it is the Cartesian product of a hypercube with either a complete graph or a hyper-octahedron. Every self-map of a pseudo-median graph that preserves or collapses edges has an invariant regular pseudo-median subgraph. Furthermore, the set of all vertices minimizing the total distance to the vertices of a pseudo-median graph induces a regular pseudo-median subgraph.     

Drawing c-planar biconnected clustered graphs

In a graph, a cluster is a set of vertices, and two clusters are said to be non-intersecting if they are disjoint or one of them is contained in the other. A clustered graph C consists of a graph G and a set of non-intersecting clusters. In this paper, we assume that C has a compound planar drawing and each cluster induces a biconnected subgraph. Then we show that such a clustered graph admits a drawing in the plane such that (i) edges are drawn as straight-line segments with no edge crossing and (ii) the boundary of the biconnected subgraph induced by each cluster is a convex polygon.     

图有[a,b]因子的邻域条件

不含有图K1，R的图称为K1,r-free图，设G是一个具有顶点集V（G）的图，设n(≥3),a和b是整数，使得b≥a≥1，若b是奇数，设b≥n-1。我们证明了每个连通的K1,r-free图G在b|V(G)|为偶数，它的最小度至少是a n-1,|V(G）≥ （2(a b)-1)(a b-1)/b，以及|NG(x）∪NG（y)|≥a|V(G)|a b对V的任意两个不邻接的点x和y都成立时，G有一个[a,b]因子。     

A forbidden subgraph characterization of line-polar bipartite graphs

A graph is polar if the vertex set can be partitioned into A and B in such a way that the subgraph induced by A is a complete multipartite graph and the subgraph induced by B is a disjoint union of cliques. Polar graphs are a common generalization of bipartite, cobipartite, and split graphs. However, recognizing polar graphs is an NP-complete problem in general. This led to the study of the polarity of special classes of graphs such as cographs and chordal graphs, cf. Ekim et al. (2008) [7] and [5]. In this paper, we study the polarity of line graphs and call a graph line-polar if its line graph is polar. We characterize line-polar bipartite graphs in terms of forbidden subgraphs. This answers a question raised in the fist reference mentioned above. Our characterization has already been used to develop a linear time algorithm for recognizing line-polar bipartite graphs, cf. Ekim (submitted for publication) [6].     

图的半强积的邻点可区别染色

两个简单图G与H的半强积G·H是具有顶点集V(G)×V(H)的简单图,其中两个顶点(u,v)与(u',v')相邻当且仅当u=u'且vv'∈E(H),或uu'∈E(G)且vv'∈E(H).图的邻点可区别边(全)染色是指相邻点具有不同色集的正常边(全)染色.统称图的邻点可区别边染色与邻点可区别全染色为图的邻点可区别染色.图G的邻点可区别染色所需的最少的颜色数称为邻点可区别染色数,并记为X_a~((r))(G),其中r=1,2,且X_a~((1))(G)与X_a~((2))(G)分别表示G的邻点可区别的边色数与全色数.给出了两个简单图的半强积的邻点可区别染色数的一个上界,并证明了该上界是可达的.然后,讨论了两个树的不同半强积具有相同邻点可区别染色数的充分必要条件.另外,确定了一类图与完全图的半强积的邻点可区别染色数的精确值.