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.     

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.     

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.     

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)     

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     

The number of vertices of degree 5 in a contraction-critically 5-connected graph

An edge of a 5-connected graph is said to be 5-contractible if the contraction of the edge results in a 5-connected graph. A 5-connected graph with no 5-contractible edge is said to be contraction-critically 5-connected. Let V(G) and V₅(G) denote the vertex set of a graph G and the set of degree 5 vertices of G, respectively. We prove that each contraction-critically 5-connected graph G has at least |V(G)|/2 vertices of degree 5. We also show that there is a sequence of contraction-critically 5-connected graphs {G_i} such that lim_i→∞|V₅(G_i)|/|V(G_i)|=1/2.     

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.     

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.     

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     

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.     

Local structure of 5- and 6-connected graphs

We prove that if graph on n vertices is minimally and contraction critically 5-connected, then it has 4n/7 vertices of degree 5. We also prove that if graph on n vertices is minimally and contraction critically 6-connected, then it has n/2 vertices of degree 6. Bibliography: 7 titles.     

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.     

A constructive characterization of contraction critical 8-connected graphs with minimum degree 9

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 graph without $k$-contractible edge is said to be contraction critically $k$-connected. Y. Egawa and W. Mader, independently, showed that the minimum degree of a contraction critical $k$-connected graph is at most $\frac{5k}{4}?1$. Hence, the minimum degree of a contraction critical 8-connected graph is either 8 or 9. This paper shows that a graph $G$ is a contraction critical 8-connected graph with minimum degree 9 if and only if $G$ is the strong product of a contraction critical 4-connected graph $H$ and $K_2$.     

Removable edges in a 5-connected graph and a construction method of 5-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [D.A. Holton, B. Jackson, A. Saito, N.C. Wormale, Removable edges in 3-connected graphs, J. Graph Theory 14(4) (1990) 465-473; H. Jiang, J. Su, Minimum degree of minimally quasi (k+1)-connected graphs, J. Math. Study 35 (2002) 187-193; T. Politof, A. Satyanarayana, Minors of quasi 4-connected graphs, Discrete Math. 126 (1994) 245-256; T. Politof, A. Satyanarayana, The structure of quasi 4-connected graphs, Discrete Math. 161 (1996) 217-228; J. Su, The number of removable edges in 3-connected graphs, J. Combin. Theory Ser. B 75(1) (1999) 74-87; J. Yin, Removable edges and constructions of 4-connected graphs, J. Systems Sci. Math. Sci. 19(4) (1999) 434-438]. In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k?5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K₆. For a k-connected graph G such that end vertices of any edge of G have at most k-3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K₆ by a number of θ⁺-operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K_k+1 or the graph H_k/2+1 obtained from K_k+2 by removing k/2+1 disjoint edges, and, if k is odd, G is isomorphic to K_k+1.     

Local structure of 7- and 8-connected graphs

We show that if graph on n vertices is minimally and contraction critically k-connected, then it has at least n/2 vertices of degree k for k = 7,8. Bibliography: 17 titles.     

Covering contractible edges in 3-connected graphs. I: Covers of size three are cutsets

An edge of a 3-connected graph is said to be contractible if its contraction results in a 3-connected graph. In this paper, a covering of contractible edges is studied. We give an alternative proof to the result of Ota and Saito (Scientia (A) 2 (1988) 101–105) that the set of contractible edges in a 3-connected graph cannot be covered by two vertices, and extended this result to a three-vertex covering. We also study the existence of a contractible edge whose contraction preserves a specified cycle, and show that a non-hamiltonian 3-connected graph has a contractible edge whose contraction preserves the circumference.     

Contractible Cliques in k-Connected Graphs

Kawarabayashi proved that for any integer k≥4, every k-connected graph contains two triangles sharing an edge, or admits a k-contractible edge, or admits a k-contractible triangle. This implies Thomassen's result that every triangle-free k-connected graph contains a k-contractible edge. In this paper, we extend Kawarabayashi's technique and prove a more general result concerning k-contractible cliques. Xingxing Yu was partially supported by NSF grant DMS-0245530 and NSA grant MDA-904-03-1-0052.     

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.     

Contractible edges in some k-connected graphs

An edge e of a k-connected graph G is said to be k-contractible (or simply contractible) if the graph obtained from G by contracting e (i.e., deleting e and identifying its ends, finally, replacing each of the resulting pairs of double edges by a single edge) is still k-connected. In 2002, Kawarabayashi proved that for any odd integer k ? 5, if G is a k-connected graph and G contains no subgraph D = K ₁ + (K ₂ ∪ K _1,2), then G has a k-contractible edge. In this paper, by generalizing this result, we prove that for any integer t ? 3 and any odd integer k ? 2t + 1, if a k-connected graph G contains neither K ₁ + (K ₂ ∪ K _1,t ), nor K ₁ + (2K ₂ ∪ K _1,2), then G has a k-contractible edge.     

Removable edges in 3-connected graphs

An edge e of a 3-connected graph G is said to be removable if G - e is a subdivision of a 3-connected graph. If e is not removable, then e is said to be nonremovable. In this paper, we study the distribution of removable edges in 3-connected graphs and prove that a 3-connected graph of order n ≥ 5 has at most [(4 n — 5)/3] nonremovable edges.