Some structural properties of minimally contraction-critically 5-connected graphs

An edge of a k-connected graph is said to be k-removable (resp. k-contractible) if the removal (resp. the contraction ) of the edge results in a k-connected graph. A k-connected graph with neither k-removable edge nor k-contractible edge is said to be minimally contraction-critically k-connected. We show that around an edge whose both end vertices have degree greater than 5 of a minimally contraction-critically 5-connected graph, there exists one of two specified configurations. Using this fact, we prove that each minimally contraction-critically 5-connected graph on n vertices has at least vertices of degree 5.     

Vertices of Degree 5 in a Contraction Critically 5-connected Graph

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 said to be contraction critically k-connected. We prove that a contraction critically 5-connected graph on n vertices has at least n/5 vertices of degree 5. We also show that, for a graph G and an integer k greater than 4, there exists a contraction critically k-connected graph which has G as its induced subgraph.     

Some properties of contraction-critical 5-connected graphs

Let k be a positive integer and let G be a k-connected graph. An edge of G is called k-contractible if its contraction still results in a k-connected graph. A non-complete k-connected graph G is called contraction-critical if G has no k-contractible edge. Let G be a contraction-critical 5-connected graph, Su proved in [J. Su, Vertices of degree 5 in contraction-critical 5-connected graphs, J. Guangxi Normal Univ. 17 (3) (1997) 12-16 (in Chinese)] that each vertex of G is adjacent to at least two vertices of degree 5, and thus G has at least vertices of degree 5. In this paper, we further study the properties of contraction-critical 5-connected graph. In the process, we investigate the structure of the subgraph induced by the vertices of degree 5 of G. As a result, we prove that a contraction-critical 5-connected graph G has at least vertices of degree 5.     

Subgraph induced by the set of degree 5 vertices in a contraction critically 5-connected graph

An edge of a 5-connected graph is said to be contractible if the contraction of the edge results in a 5-connected graph. A 5-connected graph with no contractible edge is said to be contraction critically 5-connected. Let G be a contraction critically 5-connected graph and let H be a component of the subgraph induced by the set of degree 5 vertices of G. Then it is known that |V(H)|≥4. We prove that if |V(H)|=4, then , where stands for the graph obtained from K₄ by deleting one edge. Moreover, we show that either |N_G(V(H))|=5 or |N_G(V(H))|=6 and around H there is one of two specified structures called a -configuration and a split -configuration.     

A new lower bound on the number of trivially noncontractible edges in contraction critical 5-connected graphs

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. An edge of a k-connected graph is called trivially noncontractible if its two end vertices have a common neighbor of degree k. Ando [K. Ando, Trivially noncontractible edges in a contraction critically 5-connected graph, Discrete Math. 293 (2005) 61-72] proved that a contraction critical 5-connected graph on n vertices has at least n/2 trivially noncontractible edges. Li [Xiangjun Li, Some results about the contractible edge and the domination number of graphs, Guilin, Guangxi Normal University, 2006 (in Chinese)] improved the lower bound to n+1. In this paper, the bound is improved to the statement that any contraction critical 5-connected graph on n vertices has at least trivially noncontractible edges.     

The New Lower Bound of the Number of Vertices of Degree 5 in Contraction Critical 5-Connected Graphs

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. Let G be a contraction critical 5-connected graph, in this paper we show that G has at least ${\frac{1}{2}|G|}$ vertices of degree 5.     

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     

Contractible Edges in 7-Connected Graphs

An edge of a k-connected graph is said to be a k-contractible edge, if its contraction yields again a k-connected graph. A noncomplete k-connected graph possessing no k-contractible edges is called contraction critical k-connected. Recently, Kriesell proved that every contraction critical 7-connected graph has two distinct vertices of degree 7. And he guessed that there are two vertices of degree 7 at distance one or two. In this paper, we give a proof to his conjecture. The work partially supported by NNSF of China(Grant number: 10171022)     

收缩临界5-连通图的平均度

M．Kriesell证明了收缩临界5-连通图的平均度不超过24并猜想收缩临界5-连通图的平均度小于10．本文构造了一个反例证明M．Kriesell的猜想不成立并给出了收缩临界5-连通图平均度新的上界．     

Hamiltonian 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 the following two results and conjecture that every 5-connected almost claw-free graph is hamiltonian. (1). Every 2-connected almost claw-free graph G∉J on n≤ 4 δ vertices is hamiltonian, where J is the set of all graphs defined as follows: any graph G in J can be decomposed into three disjoint connected subgraphs G ₁, G ₂ and G ₃ such that E _G (G _i , G _j ) = {u _i , u _j , v _i v _j } for i≠j and i,j = 1, 2, 3 (where u _i ≠v _i ∈V(G _i ) for i = 1, 2, 3). Moreover the bound 4δ is best possible, thereby fully generalizing several previous results. (2). Every 3-connected almost claw-free graph on at most 5δ−5 vertices is hamiltonian, hereby fully generalizing the corresponding result on claw-free graphs. Received: September 21, 1998 Final version received: August 18, 1999     

The contractible subgraph of 5-connected graphs

An edge e of a k-connected graph G is said to be k-removable if G — e is still k-connected. A subgraph H of a k-connected graph is said to be k-contractible if its contraction results still in a k-connected graph. A k-connected graph with neither removable edge nor contractible subgraph is said to be minor minimally k-connected. In this paper, we show that there is a contractible subgraph in a 5-connected graph which contains a vertex who is not contained in any triangles. Hence, every vertex of minor minimally 5-connected graph is contained in some triangle.     

Contractible Edges in k-Connected Graphs with Some Forbidden Subgraphs

In 2001, Kawarabayashi proved that for any odd integer k ≥ 3, if a k-connected graph G is \({K^{-}_{4}}\) -free, then G has a k-contractible edge. He pointed out, by a counterexample, that this result does not hold when k is even. In this paper, we have proved the following two results on the subject: (1) For any even integer k ≥ 4, if a k-connected graph G is \({K_{4}^{-}}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge. (2) Let t ≥ 3, k ≥ 2t – 1 be integers. If a k-connected graph G is \({(K_{1}+(K_{2} \cup K_{1, t}))}\) -free and d _G (x) + d _G (y) ≥ 2k + 1 hold for every two adjacent vertices x and y of V(G), then G has a k-contractible edge.     

5连通图中的可去边

An edge e of a k-connected graph G is said to be a removable edge if G O 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 NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.     

Long cycles passing through a specified path in a graph

For a graph G and an integer k ≥ 1, let ς_k(G) = d_G(v_i): {v₁, …, v_k} is an independent set of vertices in G}. Enomoto proved the following theorem. Let s ≥ 1 and let G be a (s + 2)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, ς₂(G) − s} passing through any path of length s. We generalize this result as follows. Let k ≥ 3 and s ≥ 1 and let G be a (k + s − 1)-connected graph. Then G has a cycle of length ≥ min{|V(G)|, − s} passing through any path of length s. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 177–184, 1998     

Contractible edges in minimally k-connected graphs

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. In this paper, we prove that a (K₁ + C₄)-free minimally k-connected graph has a k-contractible edge, if around each vertex of degree k, there is an edge which is not contained in a triangle. This implies previous two results, one due to Thomassen and the other due to Kawarabayashi.     

On well-covered triangulations: Part III

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper Finbow et al. (2004) [3] that there are no 5-connected planar well-covered triangulations, and in Finbow et al. (submitted for publication) [4] that there are exactly four 4-connected well-covered triangulations containing two adjacent vertices of degree 4. It is the aim of the present paper to complete the characterization of 4-connected well-covered triangulations by showing that each such graph contains two adjacent vertices of degree 4.     

Contractible edges in non-separating cycles

An edge of ak-connected graph is said to bek-contractible if the contraction of the edge results in ak-connected graph. We prove that every triangle-freek-connected graphG has an induced cycleC such that all edges ofC arek-contractible and such thatG–V(C) is (k–3)-connected (k4). This result unifies two theorems by Thomassen [5] and Egawa et. al. [3].Dedicated to Professor Toshiro Tsuzuku on his sixtieth birthday     

On the diameter of i‐center in a graph without long induced paths

A graph G is said to be P_t‐free if it does not contain an induced path on t vertices. The i‐center C_i(G) of a connected graph G is the set of vertices whose distance from any vertex in G is at most i. Denote by I(t) the set of natural numbers i, ⌊t/2⌋ ≤ i ≤ t − 2, with the property that, in every connected P_t‐free graph G, the i‐center C_i(G) of G induces a connected subgraph of G. In this article, the sharp upper bound on the diameter of G[C_i(G)] is established for every i ∈ I(t). The sharp lower bound on I(t) is obtained consequently. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 235–241, 1999     

Essential independent sets and long cycles

An independent set S of a graph G is said to be essential if S has a pair of vertices that are distance two apart in G. For SV(G) with S≠, let Δ(S)=max{d_G(x)|xS}. We prove the following theorem. Let k2 and let G be a k-connected graph. Suppose that Δ(S)d for every essential independent set S of order k. Then G has a cycle of length at least min{|G|,2d}. This generalizes a result of Fan.     

About equivalent interval colorings of weighted graphs

Given a graph G=(V,E) with strictly positive integer weights ω_i on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ω_i consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χ_int(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ω_i=1 for all vertices iV).Two k-interval colorings I₁ and I₂ are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI₁(i) if and only if π(ℓ)I₂(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.