An extension of Sallee’s theorem to infinite locally finite vap-free plane graphs

A graph isk-cyclable if givenk vertices there is a cycle that contains thek vertices. Sallee showed that every finite 3-connected planar graph is 5-cyclable. In this paper, by characterizing the circuit graphs and investigating the structure of LV-graphs, we extend his result to 3-connected infinite locally finite VAP-free plane graphs.     

Removable edges in cycles of a <Emphasis Type="Italic">k</Emphasis>-connected graph

An edge e of a k-connected graph G is said to be a removable edge if G ⊖ e is still k-connected, where G ⊖ e denotes the graph obtained from G by deleting e to get G − e, and for any end vertex of e with degree k − 1 in G − e, say x, delete x, and then add edges between any pair of non-adjacent vertices in N _G−e (x). The existence of removable edges of k-connected graphs and some properties of 3-connected graphs and 4-connected graphs have been investigated. In the present paper, we investigate some properties of k-connected graphs and study the distribution of removable edges on a cycle in a k-connected graph (k ≥ 4).     

On automorphisms of infinite graphs with forbidden subgraphs

LexX be anm-connected infinite graph without subgraphs homeomorphic toKm, n, for somen, and let α be an automorphism ofX with at least one cycle of infinite length. We characterize the structure of α and use this characterization to extend a known result about orientation-preserving automorphisms of finite plane graphs to infinite plane graphs. In the last section we investigate the action of α on the ends ofX and show that α fixes at most two ends (Theorem 3.2).     

Contractible Triples in Highly Connected Graphs

We prove that if G is k-connected (with k ≥ 2), then G contains either a cycle of length 4 or a connected subgraph of order 3 whose contraction results in a k-connected graph. This immediately implies that any k-connected graph has either a cycle of length 4 or a connected subgraph of order 3 whose deletion results in a (k − 1)-connected graph.     

The Erdős–Pósa Property for Odd Cycles in Highly Connected Graphs

Dedicated to the memory of Paul Erdős In [9] Thomassen proved that a -connected graph either contains k vertex disjoint odd cycles or an odd cycle cover containing at most 2k-2 vertices, i.e. he showed that the Erdős–Pósa property holds for odd cycles in highly connected graphs. In this paper, we will show that the above statement is still valid for 576k-connected graphs which is essentially best possible. Received November 17, 1999 RID="*" ID="*" This work was supported by a post-doctoral DONET grant. RID="†" ID="†" This work was supported by an NSF-CNRS collaborative research grant. RID="‡" ID="‡" This work was performed while both authors were visiting the LIRMM, Université de Montpellier II, France.     

Contractible edges in triangle-free graphs

An edge of a graph is calledk-contractible if the contraction of the edge results in ak-connected graph. Thomassen [5] proved that everyk-connected graph of girth at least four has ak-contractible edge. In this paper, we study the distribution ofk-contractible edges in triangle-free graphs and show the following: Whenk≧2, everyk-connected graph of girth at least four and ordern≧3k, hasn+(3/2)k ²-3k or morek-contractible edges.     

Cycles through specified vertices of a graph

We prove that ifS is a set ofk−1 vertices in ak-connected graphG, then the cycles throughS generate the cycle space ofG. Moreover, whenk≧3, each cycle ofG can be expressed as the sum of an odd number of cycles throughS. On the other hand, ifS is a set ofk vertices, these conclusions do not necessarily hold, and we characterize the exceptional cases. As corollaries, we establish the existence of odd and even cycles through specified vertices and deduce the existence of long odd and even cycles in graphs of high connectivity.     

Uniquelyk-arborable graphs

A graph is uniquelyk-arborable if its point-arboricity isk and there is only one acyclic partition of its point set intok subsets. Several properties of uniquelyk-arborable graphs are presented. One such property is that uniquelyk-arborable graphs are (k−1)-connected. Furthermore, it is shown that for any positive integerk there is a uniquelyk-arborable graph which is notk-connected.     

Mader’s Conjecture On Extremely Critical Graphs

A non-complete graph G is called an (n,k)-graph if it is n-connected but G—X is not (n−|X|+1)-connected for any X ⊂V (G) with |X|≤k. Mader conjectured that for k≥3 the graph K_2k+2−(1−factor) is the unique (2k,k)-graph(up to isomorphism). Here we prove this conjecture.     

The 3-Connected Graphs with Exactly k Non-Essential Edges

Let G be a simple 3-connected graph. An edge e of G is essential if neither the deletion G\ e nor the contraction G/e is both simple and 3-connected. In this study, we show that all 3-connected graphs with k(k ≥ 2) non-essential edges can be obtained from a wheel by three kinds of operations which defined in the paper.     

Longest Cycles in Almost Claw-Free Graphs

Some known results on claw-free graphs are generalized to the larger class of almost claw-free graphs. In this paper, we prove several properties on longest cycles in almost claw-free graphs. In particular, we show the following two results.? (1) Every 2-connected almost claw-free graph on n vertices contains a cycle of length at least min {n, 2δ+4} and the bound 2δ+ 4 is best possible, thereby fully generalizing a result of Matthews and Sumner.? (2) Every 3-connected almost claw-free graph on n vertices contains a cycle of length at least min {n, 4δ}, thereby fully generalizing a result of MingChu Li. Received: September 17, 1996 Revised: September 22, 1998     

An Algorithm to Construct <Emphasis Type="BoldItalic">k</Emphasis>-Regular <Emphasis Type="BoldItalic">k</Emphasis>-Connected Graphs with the Maximum <Emphasis Type="BoldItalic">k</Emphasis>-Diameter

 Let f(2m,k) be the Maximum k-diameter of k-regular k-connected graphs on 2m vertices. In this paper we give an algorithm and prove that we can construct k-regular k-connected graphs on 2m vertices with the maximum k-diameter using it. We also prove some known results about f(2m,k) and verify that we can get some unknown values of f(2m,k) by our algorithm. Received: December 1, 2000 Final version received: March 12, 2002 Acknowledgments. We thank the referee for many useful suggestions.     

Planarity and duality of finite and infinite graphs

We present a short proof of the following theorems simultaneously: Kuratowski's theorem, Fary's theorem, and the theorem of Tutte that every 3-connected planar graph has a convex representation. We stress the importance of Kuratowski's theorem by showing how it implies a result of Tutte on planar representations with prescribed vertices on the same facial cycle as well as the planarity criteria of Whitney, MacLane, Tutte, and Fournier (in the case of Whitney's theorem and MacLane's theorem this has already been done by Tutte). In connection with Tutte's planarity criterion in terms of non-separating cycles we give a short proof of the result of Tutte that the induced non-separating cycles in a 3-connected graph generate the cycle space. We consider each of the above-mentioned planarity criteria for infinite graphs. Specifically, we prove that Tutte's condition in terms of overlap graphs is equivalent to Kuratowski's condition, we characterize completely the infinite graphs satisfying MacLane's condition and we prove that the 3-connected locally finite ones have convex representations. We investigate when an infinite graph has a dual graph and we settle this problem completely in the locally finite case. We show by examples that Tutte's criterion involving non-separating cycles has no immediate extension to infinite graphs, but we present some analogues of that criterion for special classes of infinite graphs.     

On the connectivity of bipartite distance-balanced graphs

A connected graph Γ is said to be distance-balanced whenever for any pair of adjacent vertices u,v of Γ the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. In [K. Handa, Bipartite graphs with balanced (a,b)-partitions, Ars Combin.51 (1999), 113-119] Handa asked whether every bipartite distance-balanced graph, that is not a cycle, is 3-connected. In this paper the Handa question is answered in the negative. Moreover, we show that a minimal bipartite distance-balanced graph, that is not a cycle and is not 3-connected, has 18 vertices and is unique. In addition, we give a complete classification of non-3-connected bipartite distance-balanced graphs for which the minimal distance between two vertices in a 2-cut is three. All such graphs are regular and for each k≥3 there exists an infinite family of such graphs which are k-regular.Furthermore, we determine a number of structural properties that a bipartite distance-balanced graph, which is not 3-connected, must have. As an application, we give a positive answer to the Handa question for the subfamily of bipartite strongly distance-balanced graphs.     

On Light Graphs in 3-Connected Plane Graphs Without Triangular or Quadrangular Faces

 We prove that each 3-connected plane graph G without triangular or quadrangular faces either contains a k-path P _k , a path on k vertices, such that each of its k vertices has degree ≤5/3k in G or does not contain any k-path. We also prove that each 3-connected pentagonal plane graph G which has a k-cycle, a cycle on k vertices, k∈ {5,8,11,14}, contains a k-cycle such that all its vertices have, in G, bounded degrees. Moreover, for all integers k and m, k≥ 3, k∉ {5,8,11,14} and m≥ 3, we present a graph in which every k-cycle contains a vertex of degree at least m. Received: June 29, 1998 Final version received: April 11, 2000     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

On 4-connected 4-regular graphs without even cycle decompositions

An even cycle decomposition of a graph is a partition of its edges into even cycles. Markström constructed infinitely many 2-connected 4-regular graphs without even cycle decompositions. Má?ajová and Mazák then constructed an infinite family of 3-connected 4-regular graphs without even cycle decompositions. In this note, we further show that there exists an infinite family of 4-connected 4-regular graphs without even cycle decompositions.     

Non-separating even cycles in highly connected graphs

We prove that every k-connected graph (k ≥ 5) has an even cycle C such that G-V (C) is still (k-4)-connected. The connectivity bound is best possible.     

Graphs G for which both G and G¯ are Contraction Critically k-Connected

 An edge of a k-connected graph is said to be k-contractible if the contraction of the edge results in a k-connected graph. A k-connected graph with no k-contractible edge is called contraction critically k-connected. For k≥4, we prove that if both G and its complement Gˉ are contraction critically k-connected, then |V(G)|<k ^5/3+4k ^3/2. Received: October, 2001 Final version received: September 18, 2002 AMS Classification: 05C40     

On the Cycle Space of an Infinite 3-connected Graph

Problems related to Tutte’s theorem on the generation of the cycle space of a 3-connected finite graph are discussed for infinite graphs.