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.     

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.     

On Group Connectivity of Graphs

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero Z ₃-flow and Jaeger et al. [Group connectivity of graphs–a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165-182] further conjectured that every 5-edge-connected graph is Z ₃-connected. These two conjectures are in general open and few results are known so far. A weaker version of Tutte’s conjecture states that every 4-edge-connected graph with each edge contained in a circuit of length at most 3 admits a nowhere-zero Z ₃-flow. Devos proposed a stronger version problem by asking if every such graph is Z ₃-connected. In this paper, we first answer this later question in negative and get an infinite family of such graphs which are not Z ₃-connected. Moreover, motivated by these graphs, we prove that every 6-edge-connected graph whose edge set is an edge disjoint union of circuits of length at most 3 is Z ₃-connected. It is a partial result to Jaeger’s Z ₃-connectivity conjecture. Received: May 23, 2006. Final version received: January 13, 2008     

Group Connectivity of Bridged Graphs

Let G be a 2-edge-connected simple graph, and let A denote an abelian group with the identity element 0. If a graph G ^* is obtained by repeatedly contracting nontrivial A-connected subgraphs of G until no such a subgraph left, we say G can be A-reduced to G*. A graph G is bridged if every cycle of length at least 4 has two vertices x, y such that d _G (x, y) < d _C (x, y). In this paper, we investigate the group connectivity number Λ _g (G) = min{n: G is A-connected for every abelian group with |A| ≥ n} for bridged graphs. Our results extend the early theorems for chordal graphs by Lai (Graphs Comb 16:165–176, 2000) and Chen et al. (Ars Comb 88:217–227, 2008).     

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.     

Chromaticity of Chordal Graphs

A polynomial \(P = \lambda ^{m_0 } (\lambda - 1)^{m_1 } \cdots (\lambda - k)^{m_k } \) is said to be a chordal polynomial, if for any graph G, P(G, λ) = P implies G is a chordal graph. Using a previous result in [16], we give a necessary and sufficient condition for P to be a chordal polynomial under the condition Σm _i = k + 3.     

On b-perfect Chordal Graphs

The b-chromatic number of a graph G is the largest integer k such that G has a coloring of the vertices in k color classes such that every color class contains a vertex that has a neighbour in all other color classes. We characterize the class of chordal graphs for which the b-chromatic number is equal to the chromatic number for every induced subgraph. This research was supported by Algerian-French program CMEP/Tassili 05 MDU 639.     

Cycle Covers of Planar 2-Edge-Connected Graphs

A graph with n vertices is said to have a small cycle cover provided its edges can be covered with at most (2n ? 1)/3 cycles. Bondy [2] has conjectured that every 2-connected graph has a small cycle cover. In [3] Lai and Lai prove Bondy’s conjecture for plane triangulations. In [1] the author extends this result to all planar 3-connected graphs, by proving that they can be covered by at most (n + 1)/2 cycles. In this paper we show that Bondy’s conjecture holds for all planar 2-connected graphs. We also show that all planar 2-edge-connected graphs can be covered by at most (3n ? 3)/4 cycles and we show an infinite family of graphs for which this bound is attained.     

Matrix Completions and Chordal Graphs

In a matrix-completion problem the aim is to specifiy the missing entries of a matrix in order to produce a matrix with particular properties. In this paper we survey results concerning matrix-completion problems where we look for completions of various types for partial matrices supported on a given pattern. We see that the existence of completions of the required type often depends on the chordal properties of graphs associated with the pattern.     

Ptolemaic and Chordal Cover-Incomparability Graphs

Order - Cover-incomparability graphs (C-I graphs) are graphs whose edge-set is the union of edge-sets of the incomparability graph and the cover graph of some poset. C-I graphs captured attention...     

Retracts of Products of Chordal Graphs

In this article, we characterize the graphs G that are the retracts of Cartesian products of chordal graphs. We show that they are exactly the weakly modular graphs that do not contain K_{2, 3}, the 4‐wheel minus one spoke , and the k‐wheels (for as induced subgraphs. We also show that these graphs G are exactly the cage‐amalgamation graphs as introduced by Bre?ar and Tepeh Horvat (Cage‐amalgamation graphs, a common generalization of chordal and median graphs, Eur J Combin 30 (2009), 1071–1081); this solves the open question raised by these authors. Finally, we prove that replacing all products of cliques of G by products of Euclidean simplices, we obtain a polyhedral cell complex which, endowed with an intrinsic Euclidean metric, is a CAT(0) space. This generalizes similar results about median graphs as retracts of hypercubes (products of edges) and median graphs as 1‐skeletons of CAT(0) cubical complexes. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 161–180, 2013     

Packing r-Cliques in Weighted Chordal Graphs

In 1, we have previously observed that, in a chordal graph G, the maximum number of independent r-cliques (i.e., of vertex disjoint subgraphs of G, each isomorphic to K_r, with no edges joining any two of the subgraphs) equals the minimum number of cliques of G that meet all the r-cliques of G. When r = 1, this says that chordal graphs have independence number equal to the clique covering number. When r = 2, this is equivalent to a result of Cameron (1989), and it implies a well known forbidden subgraph characterization of split graphs. In this paper we consider a weighted version of the above min-max optimization problem. Given a chordal graph G, with a nonnegative weight for each r-clique in G, we seek a set of independent r-cliques with maximum total weight. We present two algorithms to solve this problem, based on the principle of complementary slackness. The first one operates on a graph derived from G, and is an adaptation of an algorithm of Farber (1982). The second one improves the performance by reducing the number of constraints of the linear programs. Both results produce a min-max relation. When the algorithms are specialized to the situation in which all the r-cliques have the same weight, we simplify the algorithms reported in 1, although these simpler algorithms are not as efficient. As a byproduct, we also obtain new elementary proofs of the above min-max result.     

Chordal Bipartite Graphs with High Boxicity

The boxicity of a graph G is defined as the minimum integer k such that G is an intersection graph of axis-parallel k-dimensional boxes. Chordal bipartite graphs are bipartite graphs that do not contain an induced cycle of length greater than 4. It was conjectured by Otachi, Okamoto and Yamazaki that chordal bipartite graphs have boxicity at most 2. We disprove this conjecture by exhibiting an infinite family of chordal bipartite graphs that have unbounded boxicity.     

Perfect Elimination and Chordal Bipartite Graphs

We define two types of bipartite graphs, chordal bipartite graphs and perfect elimination bipartite graphs, and prove theorems analogous to those of Dirac and Rose for chordal graphs (rigid circuit graphs, triangulated graphs). Our results are applicable to Gaussian elimination on sparse matrices where a sequence of pivots preserving zeros is sought. Our work removes the constraint imposed by Haskins and Rose that pivots must be along the main diagonal.     

Bull-Free Weakly Chordal Perfectly Orderable Graphs

 Using doubly lexical orders and the notion of box partition due to de Figueiredo, Maffray, and Porto, we show that a certain subclass of bull-free weakly chordal graphs is perfectly orderable. This together with results of de Figueiredo, Maffray, and Porto confirms Chvátal's conjecture that bull-free graphs with no anti-hole and no odd hole are perfectly orderable; here hole means induced cycle with five or more vertices. Received: September 21, 1998?Final version received: January 23, 2001     

广义乘积图的3限制边连通性

广义乘积图的概念在1984年由Bermond等人提出,Balbuena等人在2007年提出并证明了广义乘积图的边连通度和限制边连通度的上下界.继前人的工作,继续讨论证明了这类图的3限制边连通度的上下界.     

弦图的补图的最小填充

从图论观点讲,最小填充问题就是在一个图G中添加边集F,使得图G的母图G F是一个弦图而且所添边的边数| F|是最小的,其中最小值| F|称为图G的填充数,表示为f( G) .对一般图来说,最小填充问题是NP-困难的,但是对一些特殊图类来说,这个问题是在多项式时间内可解的.本文给出了弦图的补图-G的填充数f(-G) .     

n-double图的连通性

设Gl=(V1,E1),G2=(V2,E2)是两个连通图,直积(direct product)(也称为Kronecker product,tensor product和cross product) G1(×)G2的点集为V(G1(×)G2)=V(G1)(×)V(G2),边集为E(G1(×)G2)={(u1,v1)(u2,v2)∶ulu2∈E(G1),vlv2∈E(G2)}.简单图G的n-double图Dn[G]=G(×)Tn,其中n个点的全关系图Tn是完全图Kn在每个点加上一个自环得到的图.在本文中,我们研究了Dn[G]的(边)连通性,超(边)连通性.