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.     

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.     

Hamiltonicity of regular 2-connected graphs

Let G be a k-regular 2-connected graph of order n. Jackson proved that G is hamiltonian if n ≤ 3k. Zhu and Li showed that the upper bound 3k on n can be relaxed to 22/7k if G is 3-connected and k ≥ 63. We improve both results by showing that G is hamiltonian if n ≤ 7/2k − 7 and G does not belong to a restricted class F of nonhamiltonian graphs of connectivity 2. To establish this result we obtain a variation of Woodall's Hopping Lemma and use it to prove that if n ≤ 7/2k − 7 and G has a dominating cycle (i.e., a cycle such that the vertices off the cycle constitute an independent set), then G is hamiltonian. We also prove that if n ≤ 4k − 3 and G ∉ F, then G has a dominating cycle. For k ≥ 4 it is conjectured that G is hamiltonian if n ≤ 4k and G ∉ F. © 1996 John Wiley & Sons, Inc.     

Hamiltonian connectedness in 3-connected line graphs

We investigate graphs G such that the line graph L(G) is hamiltonian connected if and only if L(G) is 3-connected, and prove that if each 3-edge-cut contains an edge lying in a short cycle of G, then L(G) has the above mentioned property. Our result extends Kriesell’s recent result in [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected line graph of a claw free graph is hamiltonian connected. Another application of our main result shows that if L(G) does not have an hourglass (a graph isomorphic to K₅−E(C₄), where C₄ is an cycle of length 4 in K₅) as an induced subgraph, and if every 3-cut of L(G) is not independent, then L(G) is hamiltonian connected if and only if κ(L(G))≥3, which extends a recent result by Kriesell [M. Kriesell, All 4-connected line graphs of claw free graphs are hamiltonian-connected, J. Combin. Theory Ser. B 82 (2001) 306-315] that every 4-connected hourglass free line graph is hamiltonian connected.     

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.     

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.     

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.     

Meshes on 3-connected graphs

It is shown that in a finite 3-connected graph every pair of edges can be linked by a sequence of non-separating diagonal-free circuits C₀,…, C_n such that C_i-1 and C_i have an edge in common. Examples are given to illustrate the extent to which this theorem breaks down when the hypothesis of finiteness is removed.     

A construction of 3-connected graphs

We show that the 3-connected graphs can be generated from the complete graph on four vertices and the complete 3,3 bipartite graph by adding vertices and adding edges with endpoints on two edges meeting at a 3-valent vertex.