Colourings of oriented connected cubic graphs

In this note we show every orientation of a connected cubic graph admits an oriented 8-colouring. This lowers the best-known upper bound for the chromatic number of the family of orientations of connected cubic graphs. We further show that every such oriented graph admits a 2-dipath 7-colouring. These results imply that either the oriented chromatic number for the family of orientations of connected cubic graphs equals the 2-dipath chromatic number or the long-standing conjecture of Sopena (Sopena, 1997) regarding the chromatic number of orientations of connected cubic graphs is false.     

Decomposing planar cubic graphs

The 3‐Decomposition Conjecture states that every connected cubic graph can be decomposed into a spanning tree, a 2‐regular subgraph and a matching. We show that this conjecture holds for the class of connected plane cubic graphs.     

Generating unlabeled connected cubic planar graphs uniformly at random

We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3‐connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense‐reversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3‐connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3‐connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

PSL（2,13）的最小级连通3度弧传递陪集图表示

通过寻找给定群G的图表示,对PSL（2,13）的连通3度弧传递陪集图表示进行研究,得到了如下结果：PSL（2,13）的最小级连通3度弧传递陪集图表示的级是182.并且给出了该陪集图表示的例子.     

3限制边连通度与正则因子

设G是一个阶不小于6的k正则连通点可迁图. 如果G不含三角形, 那么图G是极大3限制边连通的, 或者G含有各连通分支都同构于同一个h阶点可迁图的k-1正则因子, 其中2k-2≤h≤3k-5. 唯一的例外是: G是围长等于4 的3正则图.     

Isomorphisms of Cubic Cayley Graphs on Dihedral Groups and Sparse Circulant Matrices

Acta Mathematica Sinica, English Series - We show that, up to isomorphism, there is a unique non-CI connected cubic Cayley graph on the dihedral group of order 2n for each even number n ≥ 4....     

Geometric representation of cubic graphs with four directions

We show that every connected cubic graph can be drawn in the plane with straight-line edges using only four distinct slopes and disconnected cubic graphs with five distinct slopes.     

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.     

Improved approximations for cubic bipartite and cubic TSP

We show improved approximation guarantees for the traveling salesman problem on cubic bipartite graphs and cubic graphs. For connected cubic bipartite graphs with n nodes, we improve on recent results of Karp and Ravi by giving a “local improvement” algorithm that finds a tour of length at most \(5/4n-2\). For 2-connected cubic graphs, we show that the techniques of Mömke and Svensson can be combined with the techniques of Correa, Larré and Soto, to obtain a tour of length at most \((4/3-1/8754)n\).     

Arc-transitive cubic Cayley graphs on PSL(2,p)

For a prime p at least 5,let T=PSL(2,p).This paper gives a classification of the connected arc-transitive cubic Cayley graphs on T and a determination of the gener- ated pairs ((?),(?)) of T such that o((?))=2 and o((?))=3.     

Dominating and total dominating partitions in cubic graphs

In this paper, we continue the study of domination and total domination in cubic graphs. It is known [Henning M.A., Southey J., A note on graphs with disjoint dominating and total dominating sets, Ars Combin., 2008, 89, 159–162] that every cubic graph has a dominating set and a total dominating set which are disjoint. In this paper we show that every connected cubic graph on nvertices has a total dominating set whose complement contains a dominating set such that the cardinality of the total dominating set is at most (n+2)/2, and this bound is essentially best possible.     

Connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups

A graph is said to be s-arc-regular if its full automorphism group acts regularly on the set of its s-arcs. In this paper, we investigate connected cubic s-arc-regular Cayley graphs of finite nonabelian simple groups. Two suffcient and necessary conditions for such graphs to be 1- or 2-arc-regular are given and based on the conditions, several infinite families of 1-or 2-arc-regular cubic Cayley graphs of alternating groups are constructed.     

Three small cubic graphs with interesting hamiltonian properties

We present here three graphs, which are the smallest known ones of their kind: a cubic three-connected planar nontraceable graph, a cubic three-connected planar graph which is not homogeneously traceable, and a cubic one-Hamiltonian graph which is not Hamiltonian connected.     

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.     

Semisymmetric cubic graphs of order 16<Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order 16p ². It is shown that for every odd prime p, there exists a semisymmetric cubic graph of order 16p ² and its structure is explicitly specified by giving the corresponding voltage rules generating the covering projections.     

Uniform cyclic edge connectivity in cubic graphs

A cubic graph which is cyclicallyk-edge connected and has the further property that every edge belongs to some cyclick-edge cut is called uniformly cyclicallyk-edge connected(U(k)). We classify theU(5) graphs and show that all cyclically 5-edge connected cubic graphs can be generated from a small finite set ofU(5) graphs by a sequence of defined operations.MATHDAHCOTAGE.AC.NZ     

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).     

Det‐extremal cubic bipartite graphs

Let G be a connected k–regular bipartite graph with bipartition V(G) = X ∪ Y and adjacency matrix A. We say G is det‐extremal if per (A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det‐extremal 3‐connected cubic bipartite graphs. We extend McCuaig's result by determining the structure of det‐extremal cubic bipartite graphs of connectivity two. We use our results to determine which numbers can occur as orders of det‐extremal connected cubic bipartite graphs, thus solving a problem due to H. Gropp. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 50–64, 2003     

Rectangular and visibility representations of infinite planar graphs

We provide a new method for extending results on finite planar graphs to the infinite case. Thus a result of Ungar on finite graphs has the following extension: Every infinite, planar, cubic, cyclically 4‐edge‐connected graph has a representation in the plane such that every edge is a horizontal or vertical straight line segment, and such that no two edges cross. A result of Tamassia and Tollis extends as follows: Every countably infinite planar graph is a subgraph of a visibility graph. Furthermore, every locally finite, 2‐connected, planar graph is a visibility graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 257–265, 2006     

A note on the smallest connected non-traceable cubic bipartite planar graph

A graph is traceable if it contains a Hamiltonian path. We present a connected non-traceable cubic bipartite planar graph with 52 vertices and prove that there are no smaller such graphs.