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。     

Cycles with Prescribed and Forbidden Sets of Elements in Cubic Graphs

 A necessary and sufficient condition for the existence of a cycle containing a set of three vertices and an edge excluding another edge is obtained for 3-connected cubic graphs by means of canonical contractions. A necessary and sufficient condition is also obtained for a cyclically 4-connected cubic graph to have a cycle that contains a set of four vertices and that avoids another vertex. Received: October 4, 1999 Final version received: June 14, 2000     

Antimagic Labeling of Cubic Graphs

An antimagic labeling of a graph G is a one‐to‐one correspondence between and such that the sum of the labels assigned to edges incident to distinct vertices are different. If G has an antimagic labeling, then we say G is antimagic. This article proves that cubic graphs are antimagic.     

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     

Minimum Power Dominating Sets of Random Cubic Graphs

We present two heuristics for finding a small power dominating set of cubic graphs. We analyze the performance of these heuristics on random cubic graphs using differential equations. In this way, we prove that the proportion of vertices in a minimum power dominating set of a random cubic graph is asymptotically almost surely at most 0.067801. We also provide a corresponding lower bound of using known results on bisection width.     

上连通和超连通的三次Bi-Cayley图

Let G be a finite group and let S(possibly, contains the identity element) be a subset of G. The Bi-Cayley graph BC(G, S) is a bipartite graph with vertex set G×{0, 1} and edge set {(g, 0) (sg, 1) : g∈G, s ∈ S}. A graph is said to be super-connected if every minimum vertex cut isolates a vertex. A graph is said to be hyper-connected if every minimum vertex cut creates two components, one of which is an isolated vertex. In this paper, super-connected and/or hyper-connected cubic Bi-Cayley graphs are characterized.     

Flows and Bisections in Cubic Graphs

A k‐weak bisection of a cubic graph G is a partition of the vertex‐set of G into two parts V₁ and V₂ of equal size, such that each connected component of the subgraph of G induced by () is a tree of at most vertices. This notion can be viewed as a relaxed version of nowhere‐zero flows, as it directly follows from old results of Jaeger that every cubic graph G with a circular nowhere‐zero r‐flow has a ‐weak bisection. In this article, we study problems related to the existence of k‐weak bisections. We believe that every cubic graph that has a perfect matching, other than the Petersen graph, admits a 4‐weak bisection and we present a family of cubic graphs with no perfect matching that do not admit such a bisection. The main result of this article is that every cubic graph admits a 5‐weak bisection. When restricted to bridgeless graphs, that result would be a consequence of the assertion of the 5‐flow Conjecture and as such it can be considered a (very small) step toward proving that assertion. However, the harder part of our proof focuses on graphs that do contain bridges.     

Independent Domination in Cubic Graphs

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by , is the minimum cardinality of an independent dominating set. In this article, we show that if is a connected cubic graph of order n that does not have a subgraph isomorphic to K_{2, 3}, then . As a consequence of our main result, we deduce Reed's important result [Combin Probab Comput 5 (1996), 277–295] that if G is a cubic graph of order n, then , where denotes the domination number of G.     

上连通和超连通三次点传递图

图G称为上连通的,若对每个最小割集C,G-C有孤立点,G称为超连通的,若对每个最小割集C,G-C恰有两个连通分支,且其中之一为弧立点,本文刻划了上连通和超连通三次点传递图。     

Longest Cycles in 3-connected Graphs with Given Independence Number

Fouquet and Jolivet conjectured that a k-connected graph of order n and independence number α ≥ k has a cycle of length at least [Fouquet and Jolivet, Problèmes combinatoires et théorie des graphes Orsay (1976), Problems, page 438]. Here we prove this conjecture for k=3.     

Petersen Cores and the Oddness of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1‐factors of G. Let be the set of edges contained in precisely i members of the k 1‐factors. Let be the smallest over all lists of k 1‐factors of G. We study lists by three 1‐factors, and call with a ‐core of G. If G is not 3‐edge‐colorable, then . In Steffen (J Graph Theory 78 (2015), 195–206) it is shown that if , then is an upper bound for the girth of G. We show that bounds the oddness of G as well. We prove that . If , then every ‐core has a very specific structure. We call these cores Petersen cores. We show that for any given oddness there is a cyclically 4‐edge‐connected cubic graph G with . On the other hand, the difference between and can be arbitrarily big. This is true even if we additionally fix the oddness. Furthermore, for every integer , there exists a bridgeless cubic graph G such that .     

长圈的邻域交条件的推广

证明了若G为一个k(k≥2)连通简单图,独立数为α,V(G)=n≥3,X1,X2,…,Xk是顶点集合V的子集,X=X1∪X2∪…∪Xk,且对于Xi(i=1,2,…,k)中任意两个不相邻点u,v,|N(u)∩N(v)|≥α,则X在G中可圈,并给出几个相关推论.     

强竞赛图中包含一条路的圈

设T为强竞赛图,P为T中一条长度为2的路.给出了T存在包含P的一些圈的充分条件.     

Nowhere‐Zero 5‐Flows On Cubic Graphs with Oddness 4

Tutte's 5‐flow conjecture from 1954 states that every bridgeless graph has a nowhere‐zero 5‐flow. It suffices to prove the conjecture for cyclically 6‐edge‐connected cubic graphs. We prove that every cyclically 6‐edge‐connected cubic graph with oddness at most 4 has a nowhere‐zero 5‐flow. This implies that every minimum counterexample to the 5‐flow conjecture has oddness at least 6.     

Construction of Hamiltonian Cycles in Layered Cubic Planar Graphs

 In this paper, a class of cubic planar graphs is given that have Hamiltonian cycles that can be constructed in linear time. A member of this class is called a layered cubic planar graph, and consists of a sequence of cycles C ₀ ,C ₁ ,…,C _n such that each pair of successive cycles, C _i , C _i+1 , is joined by a matching. The cycles can be pictured as concentric circles, and the edges of the matchings as radial line segments between successive circles. The subgraph bounded by two successive cycles forms a layer; each face in layer i is incident to a fixed number k _i+1 of edges in the matching in layer i+1. The problem that initially motivated this work is that of identifying classes of convex cubic polyhedra that can be easily edge three-colored. Received: September 21, 1998 Final version received: July 21, 1999     

On Cubic Bridgeless Graphs Whose Edge‐Set Cannot be Covered by Four Perfect Matchings

The problem of establishing the number of perfect matchings necessary to cover the edge‐set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this article, we prove that deciding whether this number is at most four for a given cubic bridgeless graph is NP‐complete. We also construct an infinite family of snarks (cyclically 4‐edge‐connected cubic graphs of girth at least 5 and chromatic index 4) whose edge‐set cannot be covered by four perfect matchings. Only two such graphs were known. It turns out that the family also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with m edges has length at least , and we show that this inequality is strict for graphs of . We also construct the first known snark with no cycle cover of length less than .     

On Cubic Graphs Admitting an Edge-Transitive Solvable Group

Using covering graph techniques, a structural result about connected cubic simple graphs admitting an edge-transitive solvable group of automorphisms is proved. This implies, among other, that every such graph can be obtained from either the 3-dipole Dip₃ or the complete graph K ₄, by a sequence of elementary-abelian covers. Another consequence of the main structural result is that the action of an arc-transitive solvable group on a connected cubic simple graph is at most 3-arc-transitive. As an application, a new infinite family of semisymmetric cubic graphs, arising as regular elementary abelian covering projections of K _3,3, is constructed.     

On Compatible Normal Odd Partitions in Cubic Graphs

A normal odd partition of the edges of a cubic graph is a partition into trails of odd length (no repeated edge) such that each vertex is the end vertex of exactly one trail of the partition and internal in some trail. For each vertex v, we can distinguish the edge for which this vertex is pending. Three normal odd partitions are compatible whenever these distinguished edges are distinct for each vertex. We examine this notion and show that a cubic 3‐edge‐colorable graph can always be provided with three compatible normal odd partitions. The Petersen graph has this property and we can construct other cubic graphs with chromatic index four with the same property. Finally, we propose a new conjecture which, if true, would imply the well‐known Fan and Raspaud Conjecture.     

具有三次曲线解xy2+y=x3的中心对称三次系统极限环的存在性

本文证明了具有三次曲线解ｘｙ²＋ｙ＝ｘ³的中心对称三次系统的极限环存在，而且至少可以存在四个极限环，它们作（２，２）分布．从而纠正了文［１］的结论     

Cycles Containing a Given Arc in Regular Multipartite Tournaments

In this paper we prove that if T is a regular n-partite tournament with n≥>6, then each arc of T lies on a k-cycle for k=4,5,…,n. Our result generalizes theorems due to Alspach and Guo respectively.