Paired-Domination in Claw-Free Cubic Graphs

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The minimum cardinality of a paired-dominating set of G is the paired-domination number of G, denoted by _pr(G). If G does not contain a graph F as an induced subgraph, then G is said to be F-free. In particular if F=K_1,3 or K₄–e, then we say that G is claw-free or diamond-free, respectively. Let G be a connected cubic graph of order n. We show that (i) if G is (K_1,3,K₄–e,C₄)-free, then _pr(G)3n/8; (ii) if G is claw-free and diamond-free, then _pr(G)2n/5; (iii) if G is claw-free, then _pr(G)n/2. In all three cases, the extremal graphs are characterized.Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal. This paper was written while the second author was visiting the Laboratoire de Recherche en Informatique (LRI) at the Université de Paris-Sud in July 2002. The second author thanks the LRI for their warm hospitality     

The b-Chromatic Number of Cubic Graphs

The b-chromatic number of a graph G is the largest integer k such that G admits a proper k-coloring in which every color class contains at least one vertex adjacent to some vertex in all the other color classes. It is proved that with four exceptions, the b-chromatic number of cubic graphs is 4. The exceptions are the Petersen graph, K _3,3, the prism over K ₃, and one more sporadic example on 10 vertices.     

4p阶三度点传递图

一个图称为点传递图或对称图如果它的自同构群分别在点集或点集有序对上传递.设P为素数,给出了4p阶连通三度点传递图分类(徐明曜等在[Chin.Ann.Math.,2004,25B(4):545-554]中分类了4p阶连通三度对称图).确定了4p阶互不同构的连通三度点传递图的个数f(4p);当P=2,3,5,7时,f(4p)分别为2,4,8,6;当P≥11且4|(p-1)时,f(4p)=5+p-3/2,当P≥11且4|(p-1)时,f(4p)=3+p-3/2.     

Paired-Domination Subdivision Numbers of Graphs

A paired-dominating set of a graph G = (V, E) with no isolated vertex is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G, denoted by γ _pr (G), is the minimum cardinality of a paired-dominating set of G. The paired-domination subdivision number sd _γpr (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the paired-domination number. In this paper we establish upper bounds on the paired-domination subdivision number and pose some problems and conjectures.     

Spectral Characterization of Some Cubic Graphs

It is proved that the Cartesian product of an odd cycle with the complete graph on 2 vertices, is determined by the spectrum of the adjacency matrix. We also present some computational results on the spectral characterization of cubic graphs on at most 20 vertices.     

2p2阶3度点传递图

一个图称为点传递图,如果它的全自同构群在它的顶点集合上作用传递.本文证明了一个2p～2（p为素数）阶连通3度点传递图或者是Calyley图,或者同构于广义Petersen图P（p～2,t）,这里t～2≡-1（modp～2）.     

Classifying Cubic Semisymmetric Graphs of Order 18 p n

A graph X is said to be G-semisymmetric if it is regular and there exists a subgroup G of A := Aut (X) acting transitively on its edge set but not on its vertex set. In the case of G = A, we call X a semisymmetric graph. Let p be a prime. It was shown by Folkman (J Comb Theory 3:215–232, 1967) that a regular edge-transitive graph of order 2p or 2p ² is necessarily vertex-transitive. The smallest semisymmetric graph is the Folkman graph. In this study, we classify all connected cubic semisymmetric graphs of order 18p ⁿ , where p is a prime and \({n \geq 1}\) .     

关于4p阶3度Cayley图的正规性的一点注记

群G的Cayley图Cay(G,S)称为是正规的,如果G的右正则表示R(G)在Cay(G,S)的全自同构群中正规.设p为奇素数,相关文献决定了4p阶连通3度Cayley图的正规性.本文给出了上述文献的主要结果的一个新的简短的证明.     

Upper Bounds for the Paired-Domination Numbers of Graphs

A set \(S\subseteq V\) is a paired-dominating set if every vertex in \(V{\setminus } S\) has at least one neighbor in S and the subgraph induced by S contains a perfect matching. The paired-domination number of a graph G, denoted by \(\gamma _{pr}(G)\), is the minimum cardinality of a paired-dominating set of G. A conjecture of Goddard and Henning says that if G is not the Petersen graph and is a connected graph of order n with minimum degree \(\delta (G)\ge 3\), then \(\gamma _{pr}(G)\le 4n/7\). In this paper, we confirm this conjecture for k-regular graphs with \(k\ge 4\).     

Characterization of Product Anti-Magic Graphs of Large Order

A graph G is product anti-magic if one can bijectively label its edges with integers 1, . . . ,e(G) so that no two vertices have the same product of incident labels. This property was introduced by Figueroa-Centeno, Ichishima, and Muntaner-Batle who in particular conjectured that every connected graph with at least 4 vertices is product anti-magic. Here, we completely describe all product anti-magic graphs of sufficiently large order, confirming the above conjecture in this case. Our proof uses probabilistic methods. Reverts to public domain 28 years from publication. Partially supported by the National Science Foundation, Grant DMS-0457512.     

A Characterization of Graphs Where the Independence Number Equals the Radius

In a classical 1986 paper by Erdös, Saks and Saós every graph of radius r has an induced path of order at least 2r ? 1. This result implies that the independence number of such graphs is at least r. In this paper, we use a result of S. Fajtlowicz about radius-critical graphs to characterize graphs where the independence number is equal to the radius, for all possible values of the radius except 2, 3, and 4. We briefly discuss these remaining cases as well.     

高度平面图完备色数的一个刻划

设G是一个平面图,△(G)为G的最大度．G的完备色数x     

原始图与三次图的可圈度

本文综述关于原始图与三次图的可圈度的近期结果并提出一些未解决的问题。     

A Note on Groups Associated with 4-Arc-Transitive Cubic Graphs

A cubic (trivalent) graph is said to be 4-arc-transitive ifits automorphism group acts transitively on the 4-arcs of (wherea 4-arc is a sequence v₀, v₁, ... v₄ of vertices of such thatv_i–1 is adjacent to v_i for 1 i 4, and v_i–1 v_i+1for 1 i < 4). In his investigations into graphs of thissort, Biggs defined a family of groups 4⁺(a^m), for m = 3,4,5...,each presented in terms of generators and relations under theadditional assumption that the vertices of a circuit of lengthm are cyclically permuted by some automorphism. In this paperit is shown that whenever m is a proper multiple of 6, the group4⁺(a^m) is infinite. The proof is obtained by constructing transitivepermutation representations of arbitrarily large degree.     

Bicliques in Graphs I: Bounds on Their Number

Bicliques are inclusion-maximal induced complete bipartite subgraphs in graphs. Upper bounds on the number of bicliques in bipartite graphs and general graphs are given. Then those classes of graphs where the number of bicliques is polynomial in the vertex number are characterized, provided the class is closed under induced subgraphs. Received January 27, 1997     

An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs

In this paper, we continue the study of total restrained domination in graphs. A set S of vertices in a graph G = (V, E) is a total restrained dominating set of G if every vertex of G is adjacent to some vertex in S and every vertex of ${V {\setminus} S}$ is adjacent to a vertex in ${V {\setminus} S}$ . The minimum cardinality of a total restrained dominating set of G is the total restrained domination number γ _tr(G) of G. Jiang et?al. (Graphs Combin 25:341–350, 2009) showed that if G is a connected cubic graph of order n, then γ _tr(G) ≤ 13n/19. In this paper we improve this upper bound to γ _tr(G) ≤ (n?+?4)/2. We provide two infinite families of connected cubic graphs G with γ _tr(G) = n/2, showing that our new improved bound is essentially best possible.     

单圈图的最大特征值序

主要讨论了单圈图按其最大特征值进行排序的问题，确定了该序的前六个图。     

Amalgams of Cubic Bipartite Graphs

The amalgamation technique has been introduced for groups by Higman et al. [8] and Goldschmidt [7] and developed on geometries by Kegel and Schleiermacher [9]. We define a “graph amalgam” to point out a different approach to certain classes of cubic bipartite graphs. Furthermore, we find relations between graph amalgams, 3-bridges and star-products of cubic bipartite graphs.     

A Linear Heterochromatic Number of Graphs

Let G=(V(G),E(G)) be a multigraph with multiple loops allowed, and V ₀V(G). We define h(G,V ₀) to be the minimum integer k such that for every edge-colouring of G using exactly k colours, all the edges incident with some vertex in V ₀ receive different colours. In this paper we prove that if each xV ₀ is incident to at least two edges of G, then h(G,V ₀)=(G[V ₀])+|E(G)|–|V ₀|+1 where (G[V ₀]) is the maximum cardinality of a set of mutually disjoint cycles (of length at least two) in the subgraph induced by V ₀. Acknowledgments.We thank the referee for suggesting us a short alternative proof of our main theorem.