On 3-Edge-Connected Supereulerian Graphs

The supereulerian graph problem, raised by Boesch et al. (J Graph Theory 1:79–84, 1977), asks when a graph has a spanning eulerian subgraph. Pulleyblank showed that such a decision problem, even when restricted to planar graphs, is NP-complete. Jaeger and Catlin independently showed that every 4-edge-connected graph has a spanning eulerian subgraph. In 1992, Zhan showed that every 3-edge-connected, essentially 7-edge-connected graph has a spanning eulerian subgraph. It was conjectured in 1995 that every 3-edge-connected, essentially 5-edge-connected graph has a spanning eulerian subgraph. In this paper, we show that if G is a 3-edge-connected, essentially 4-edge-connected graph and if for every pair of adjacent vertices u and v, d _G (u) + d _G (v) ≥ 9, then G has a spanning eulerian subgraph.     

Cycle Double Covers in Cubic Graphs having Special Structures

In the first part of this article, we employ Thomason's Lollipop Lemma 25 to prove that bridgeless cubic graphs containing a spanning lollipop admit a cycle double cover (CDC) containing the circuit in the lollipop; this implies, in particular, that bridgeless cubic graphs with a 2‐factor F having two components admit CDCs containing any of the components in the 2‐factor, although it need not have a CDC containing all of F. As another example consider a cubic bridgeless graph containing a 2‐factor with three components, all induced circuits. In this case, two of the components may separately be used to start a CDC although it is uncertain whether the third component may be part of some CDC. Numerous other corollaries shall be given as well. In the second part of the article, we consider special types of bridgeless cubic graphs for which a prominent circuit can be shown to be included in a CDC. The interest here is the proof technique and therefore we only give the simplest case of the theorem. Notably, we show that a cubic graph that consists of an induced 2k‐circuit C together with an induced 4k‐circuit T and an independent set of 2k vertices, each joined by one edge to C and two edges to T, has a CDC starting with T.     

Group Connectivity in 3-Edge-Connected Graphs

Let A be an abelian group with |A|?≥ 4. For integers k and l with k?>?0 and l?≥ 0, let ${{\mathcal C}(k, l)}$ denote the family of 2-edge-connected graphs G such that for each edge cut ${S\subseteq E(G)}$ with two or three edges, each component of G ? S has at least (|V(G)| ? l)/k vertices. In this paper, we show that if G is 3-edge-connected and ${G\in {\mathcal C}(6,5)}$ , then G is not A-connected if and only if G can be A-reduced to the Petersen graph.     

Group Connectivity of 3-Edge-Connected Chordal Graphs

Let A be a finite abelian group and G be a digraph. The boundary of a function f: E(G)ZA is a function ‘f: V(G)ZA given by ‘f(v)=~_{e leaving v}f(e)m~_{e entering v}f(e). The graph G is A-connected if for every b: V(G)ZA with ~_{v] V(G)} b(v)=0, there is a function f: E(G)ZA{0} such that ‘f=b. In [J. Combinatorial Theory, Ser. B 56 (1992) 165-182], Jaeger et al showed that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̈́. It is conjectured that every 3-edge-connected graph is A-connected, for every abelian group A with |A|̓ and that every 5-edge-connected graph is A-connected, for every abelian group A with |A|́.¶ In this note, we investigate the group connectivity of 3-edge-connected chordal graphs and characterize 3-edge-connected chordal graphs that are A-connected for every finite abelian group A with |A|́.     

1‐Factor and Cycle Covers 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. Any list of three 1‐factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge‐covers and for the existence of three 1‐factors with empty intersection. Furthermore, if , then is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with have a 4‐cycle cover of length and a 5‐cycle double cover. These graphs also satisfy two conjectures of Zhang 18 . We also give a negative answer to a problem stated in 18 .     

Almost Covers Of 2-Arc Transitive Graphs

Let be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition with block size v. Let be the quotient graph of relative to and [B,C] the bipartite subgraph of induced by adjacent blocks B,C of . In this paper we study such graphs for which is connected, (G, 2)-arc transitive and is almost covered by in the sense that [B,C] is a matching of v-1 2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case K _v+1 is covered by results of Gardiner and Praeger. We consider here the general case where K _v+1, and prove that, for some even integer n 4, is a near n-gonal graph with respect to a certain G-orbit on n-cycles of . Moreover, we prove that every (G, 2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient of a graph with these properties. (A near n-gonal graph is a connected graph of girth at least 4 together with a set of n-cycles of such that each 2-arc of is contained in a unique member of .)     

三边连通超欧拉图的一个注记

For two integers l 0 and k ≥ 0,define C(l,k) to be the family of 2-edge connected graphs such that a graph G ∈ C(l,k) if and only if for every bond S-E(G) with |S| ≤ 3,each component of G-S has order at least(|V(G)|-k)/l.In this note we prove that if a 3-edge-connected simple graph G is in C(10,3),then G is supereulerian if and only if G cannot be contracted to the Petersen graph.Our result extends an earlier result in [Supereulerian graphs and Petersen graph.JCMCC 1991,9:79-89] by Chen.     

Covers of Point-Hyperplane Graphs

A cover of the non-incident point-hyperplane graph of projective dimension 3 for fields of characteristic 2 is constructed. For fields of even order larger than 2, this leads to an elementary construction of the non-split extension of SL₄( )by ⁶.     

Circuit Covers of Signed Graphs

We introduce the concept of a signed circuit cover of a signed graph. A signed circuit cover is a natural analog of a circuit cover of a graph and is equivalent to a covering of the corresponding signed graphic matroid with circuits. As in the case of graphs, a signed graph has a signed circuit cover only when it admits a nowhere‐zero integer flow. In the present article, we establish the existence of a universal coefficient such that every signed graph G that admits a nowhere‐zero integer flow has a signed circuit cover of total length at most . We show that if G is bridgeless, then , and in the general case .     

Elementary Abelian Covers of Graphs

Let _G (X) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G Aut X of automorphisms lifts. There is a natural lattice structure on _G (X), where _{1 2} whenever ₂ factors through ₁. The sublattice _G () of coverings which are below a given covering : X^~ X naturally corresponds to a lattice _G () of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications.For a given prime p, let _G ^p (X) _G (X) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs _G ^p (X) in the sense that, for each member of _G ^p (X), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.     

Oriented Coloring of Triangle-Free Planar Graphs and 2-Outerplanar Graphs

A graph is planar if it can be embedded on the plane without edge-crossings. A graph is 2-outerplanar if it has a planar embedding such that the subgraph obtained by removing the vertices of the external face is outerplanar (i.e. with all its vertices on the external face). An oriented k-coloring of an oriented graph G is a homomorphism from G to an oriented graph H of order k. We prove that every oriented triangle-free planar graph has an oriented chromatic number at most 40, that improves the previous known bound of 47 [Borodin, O. V. and Ivanova, A. O., An oriented colouring of planar graphs with girth at least 4, Sib. Electron. Math. Reports, vol. 2, 239–249, 2005]. We also prove that every oriented 2-outerplanar graph has an oriented chromatic number at most 40, that improves the previous known bound of 67 [Esperet, L. and Ochem, P. Oriented colouring of 2-outerplanar graphs, Inform. Process. Lett., vol. 101(5), 215–219, 2007].     

The Linear 2-Arboricity of Planar Graphs

 Let G be a planar graph with maximum degree Δ and girth g. The linear 2-arboricity la ₂(G) of G is the least integer k such that G can be partitioned into k edge-disjoint forests, whose component trees are paths of length at most 2. We prove that (1) la ₂(G)≤⌈(Δ+1)/2⌉+12; (2) la ₂(G)≤⌈(Δ+1)/2⌉+6 if g≥4; (3) la ₂(G)≤⌈(Δ+1)/2⌉+2 if g≥5; (4) la ₂(G)≤⌈(Δ+1)/2⌉+1 if g≥7. Received: June 28, 2001 Final version received: May 17, 2002 Acknowledgments. This work was done while the second and third authors were visiting the Institute of Mathematics, Academia Sinica, Taipei. The financial support provided by the Institute is greatly appreciated.     

On Orthogonal Double Covers of Graphs

A transformation which allows us to obtain an orthogonal double cover of a graph G from any permutation of the edge set of G is described. This transformation is used together with existence results for self-orthogonal latin squares, to give a simple proof of a conjecture of Chung and West.     

Shortest Circuit Covers of Cubic Graphs

We show that the edge set of a bridgeless cubic graph G can be covered with circuits such that the sum of the lengths of the circuits is at most |E(G)|. Stronger results are obtained for cubic graphs of large girth.     

On Orthogonal Double Covers of Graphs

An orthogonal double cover (ODC) is a collection of n spanning subgraphs(pages) of the complete graph K _n such that they cover every edge of the completegraph twice and the intersection of any two of them contains exactly one edge. If all the pages are isomorphic tosome graph G, we speak of an ODC by G. ODCs have been studied for almost 25 years, and existenceresults have been derived for many graph classes. We present an overview of the current state of research alongwith some new results and generalizations. As will be obvious, progress made in the last 10 years is in many waysrelated to the work of Ron Mullin. So it is natural and with pleasure that we dedicate this article to Ron, on theoccasion of his 65th birthday.     

Strong 5-Cycle Double Covers of Graphs

The Cycle Double Cover Conjecture claims that every bridgeless graph has a cycle double cover and the Strong Cycle Double Conjecture states that every such graph has a cycle double cover containing any specified circuit. In this paper, we get a necessary and sufficient condition for bridgeless graphs to have a strong 5-cycle double cover. Similar condition for the existence of 5-cycle double covers is also obtained. These conditions strengthen/improve some known results.     

Random Walks on Directed Covers of Graphs

Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still equals the branching number, upper and lower growth rates no longer coincide in general. Furthermore, the behavior of random walks on directed covers of infinite graphs is more subtle. We provide a classification in terms of recurrence and transience and point out that the critical random walk may be recurrent or transient. Our proof is based on the observation that recurrence of the random walk is equivalent to the almost sure extinction of an appropriate branching process. Two examples in random environment are provided: homesick random walk on infinite percolation clusters and random walk in random environment on directed covers. Furthermore, we calculate, under reasonable assumptions, the rate of escape with respect to suitable length functions and prove the existence of the asymptotic entropy providing an explicit formula which is also a new result for directed covers of finite graphs. In particular, the asymptotic entropy of random walks on directed covers of finite graphs is positive if and only if the random walk is transient.     

Small Universal Graphs for Bounded-Degree Planar Graphs

For all positive integers N and k, let denote the family of planar graphs on N or fewer vertices, and with maximum degree k. For all positive integers N and k, we construct a -universal graph of size . This construction answers with an explicit construction the previously open question of the existence of such a graph. Received July 8, 1998 RID="*" ID="*" Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.