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     

立方图的对控制数

设G=(V,E)是一个简单图, 对任意的顶点子集合 $S\subseteq V$, G[S]表示图G中由S所导出的子图. 如果S是G的一个控制集并且G[S]包含至少一个完备匹配, 则称S是G的一个对控制集. G中对控制集的最少的顶点数称为$G$的对控制数, 记为γ_p(G). 该文证明了对任意有n点的连通立方图G, γ_p(G)≤3n/ 5.     

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.     

无爪3 -正则图的独立数

如果图G的一个集合X中任两个点不相邻, 则称 X 为独立集合. 如果 N[X]=V(G), 则称X是一个控制集合. i(G)(β(G))分别表示所有极大独立集合的最小(最大)基数. γ(G)(Γ(G))表示所有极小控制集合的最小(最大)基数. 在这篇论文中, 作者证明如下结论: (1) 如果 G ∈R 且G 是n阶3 -正则图, 则 γ(G)= i(G), β(G)=n/3. (2) 每个n阶连通无爪3 -正则图 G, 如果 G(G≠ K₄) 且不含诱导子图K₄-e, 则 β(G) =n/3.     

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.     

4p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.证明了一个4p(p为素数)阶连通3度点传递图或者是Cayley图,或者同构于下列之一;广义Petersen图P(10,2),正十二面体,Coxeter图,或广义Petersen图P(2p,k),这里k2≡-1(mod 2p).     

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）.     

Cubic Semisymmetric Graphs of Order 2qp2

Acta Mathematicae Applicatae Sinica, English Series - A regular edge-transitive graph is said to be semisymmetric if it is not vertex-transitive. Let p be a prime. By Folkman [J. Combin. Theory 3...     

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 \({n \geq 1}\) .     

具有最大控制数的连通图的刻画

设Ｇ为一个Ｐ阶图,γ（Ｇ）表示Ｇ的控制数．显然γ（Ｇ）≤［ｐ／２］．本文的目的是刻画达到这个上界的连通图．主要结果：（１）当ｐ为偶数时,γ（Ｇ）＝ｐ／２当且仅当Ｇ≈C₄或者Ｇ为某连通图的冠;（２）当ｐ为奇数时,γ（Ｇ）＝(p-1)/2当且仅当Ｇ的每棵生成树为定理３．１中所示的两类树之一．     

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 \(k\ge 4\).     

Vertex‐Transitive Cubic Graphs of Square‐Free Order

A classification of connected vertex‐transitive cubic graphs of square‐free order is provided. It is shown that such graphs are well‐characterized metacirculants (including dihedrants, generalized Petersen graphs, Möbius bands), or Tutte's 8‐cage, or graphs arisen from simple groups PSL(2, p).     

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     

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.