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.     

5-Shredders in 5-connected graphs

For a graph G, a subset S of V(G) is called a shredder if G−S consists of three or more components. We show that if G is a 5-connected graph with |V(G)|≥135, then the number of shredders of cardinality 5 of G is less than or equal to (2|V(G)|−10)/3.     

Hamiltonicity of 6-connected line graphs

Let G be a graph and let D₆(G)={v∈V(G)|d_G(v)=6}. In this paper we prove that: (i) If G is a 6-connected claw-free graph and if |D₆(G)|≤74 or G[D₆(G)] contains at most 8 vertex disjoint K₄’s, then G is Hamiltonian; (ii) If G is a 6-connected line graph and if |D₆(G)|≤54 or G[D₆(G)] contains at most 5 vertex disjoint K₄’s, then G is Hamilton-connected.     

Generating thec *5-connected graphs

A planar 3-valent 3-connected graph is said to becyclically n-connected provided it is possible to separate two circuits by cutting edges, and at leastn edges must be cut to do so. The graph is said to bestrongly cyclically n-connected provided it is cyclicallyn-connected and any separation of two circuits by cutting edges leaves one component consisting of just a simple circuit. We give a method of generating all strongly cyclically 5-connected graphs by certain types of facet splitting.     

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.     

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.     

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.     

Removable edges of cycles in 5-connected graphs

Let G be a 5-connected graph. For an edge e of G, we do the following operations on G: first, delete the edge e from G, resulting the graph G?e; second, for each vertex x of degree 4 in G?e, delete x from G?e and then completely connect the 4 neighbors of x by K ₄. If multiple edges occur, we use single edge to replace them. The final resultant graph is denoted by G ? e. If G ? e is still 5-connected, then e is called a removable edge of G. In this paper, we investigate the distribution of removable edges in a cycle of a 5-connected graph. And we give examples to show some of our results are best possible in some sense.     

On -connected graphs

A graph is called -connected if is -edge-connected and is -edge-connected for every vertex . The study of -connected graphs is motivated by a theorem of Thomassen [J. Combin. Theory Ser. A 110 (2015), pp. 67–78] (that was a conjecture of Frank [SIAM J. Discrete Math. 5 (1992), no. 1, pp. 25–53]), which states that a graph has a -vertex-connected orientation if and only if it is (2,2)-connected. In this paper, we provide a construction of the family of -connected graphs for even, which generalizes the construction given by Jordán [J. Graph Theory 52 (2006), pp. 217–229] for (2,2)-connected graphs. We also solve the corresponding connectivity augmentation problem: given a graph and an integer , what is the minimum number of edges to be added to make -connected. Both these results are based on a new splitting-off theorem for -connected graphs.     

Pairs of Hamiltonian circuits in 5-connected planar graphs

Settling a problem raised by B. Grünbaum, J. Malkevitch, and the author, we present 5-valent 5-connected planar graphs that admit no pairs of edgedisjoint Hamiltonian circuits; our smallest example has 176 vertices. This is used to construct an infinite family of 5-valent 5-connected planar graphs, in which every member has the property that any pair of Hamiltonian circuits in it share at least about $\text{1}\text{168}$ of their edges. We construct 4- and 5-valent, 3-connected non-Hamiltonian planar graphs.     

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.     

Bounding the number of embeddings of 5-connected projective-planar graphs

A graph is said to be projective-planar if it is nonplanar and is embeddable in a projective plane. In this paper we show that the numbers of projectiveplanar embeddings (up to equivalence) of all 5-connected graphs have an upper bound c(?120).     

On 3-connected minors of 3-connected matroids and graphs

Whittle proved, for k=1,2, that if N is a 3-connected minor of a 3-connected matroid M, satisfying r(M)−r(N)≥k, then there is a k-independent set I of M such that, for every x∈I, si(M/x) is a 3-connected matroid with an N-minor. In this paper, we establish this result for k=3. It is already known that it cannot be extended to greater values of k. But, here we also show that, in the graphic case, with the extra assumption that r(M)−r(N)≥6, we can guarantee the existence of a 4-independent set of M with such a property. Moreover, in the binary case, we show that if r(M)−r(N)≥5, then M has such a 4-independent set or M has a triangle T meeting 3 triads and such that M/T is a 3-connected matroid with an N-minor.     

Uncontractable 4-connected graphs

The only uncontractable 4-connected graphs are C²_n for n ≥ 5 and the line graphs of the cubic cyclically 4-connected graphs.     

Minimally 3-connected graphs

A constructive characterization of the class of minimally 3-connected graphs is presented. This yields a new characterization for the class of 3-connected graphs, which differs from the characterization provided by Tutte. Where Tutte's characterization requires the set of all wheels as a starting set, the new characterization requires only the graph K₄. The new characterization is based on the application of graph operations to appropriate vertex and edge sets in minimally 3-connected graphs.     

Uniformly 3-connected graphs

Let $k$ be a positive integer. A graph is said to be uniformly $k$-connected if between any pair of vertices the maximum number of independent paths is exactly $k$. Dawes showed that all minimally 3-connected graphs can be constructed from $K_4$ such that every graph in each intermediate step is also minimally 3-connected. In this paper, we generalize Dawes' result to uniformly 3-connected graphs. We give a constructive characterization of the class of uniformly 3-connected graphs which differs from the characterization provided by Göring et al., where their characterization requires the set of all 3-connected and 3-regular graphs as a starting set, the new characterization requires only the graph $K_4$. Eventually, we obtain a tight bound on the number of edges in uniformly 3-connected graphs.     

Decomposition of 3-connected graphs

Cunningham and Edmonds [4[ have proved that a 2-connected graphG has a unique minimal decomposition into graphs, each of which is either 3-connected, a bond or a polygon. They define the notion of a good split, and first prove thatG has a unique minimal decomposition into graphs, none of which has a good split, and second prove that the graphs that do not have a good split are precisely 3-connected graphs, bonds and polygons. This paper provides an analogue of the first result above for 3-connected graphs, and an analogue of the second for minimally 3-connected graphs. Following the basic strategy of Cunningham and Edmonds, an appropriate notion of good split is defined. The first main result is that ifG is a 3-connected graph, thenG has a unique minimal decomposition into graphs, none of which has a good split. The second main result is that the minimally 3-connected graphs that do not have a good split are precisely cyclically 4-connected graphs, twirls (K _3,n for somen3) and wheels. From this it is shown that ifG is a minimally 3-connected graph, thenG has a unique minimal decomposition into graphs, each of which is either cyclically 4-connected, a twirl or a wheel.Research partially supported by Office of Naval Research Grant N00014-86-K-0689 at Purdue University.     

Hamiltonicity of 4-connected graphs

Let σk（G） denote the minimum degree sum of k independent vertices in G and α（G） denote the number of the vertices of a maximum independent set of G. In this paper we prove that if G is a 4-connected graph of order n and σ5（G） 〉 n ＋ 3σ（G） ＋ 11, then G is Hamiltonian.