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.     

Hourglasses and Hamilton cycles in 4‐connected claw‐free graphs

We show that if G is a 4‐connected claw‐free graph in which every induced hourglass subgraph S contains two non‐adjacent vertices with a common neighbor outside S, then G is hamiltonian. This extends the fact that 4‐connected claw‐free, hourglass‐free graphs are hamiltonian, thus proving a broader special case of a conjecture by Matthews and Sumner. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 267–276, 2005     

Prism‐hamiltonicity of triangulations

The prism over a graph G is the Cartesian product G □ K₂ of G with the complete graph K₂. If the prism over G is hamiltonian, we say that G is prism‐hamiltonian. We prove that triangulations of the plane, projective plane, torus, and Klein bottle are prism‐hamiltonian. We additionally show that every 4‐connected triangulation of a surface with sufficiently large representativity is prism‐hamiltonian, and that every 3‐connected planar bipartite graph is prism‐hamiltonian. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 181–197, 2008     

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.     

Maximal K3's and Hamiltonicity of 4‐connected claw‐free graphs

Let cl(G) denote Ryjá?ek's closure of a claw‐free graph G. In this article, we prove the following result. Let G be a 4‐connected claw‐free graph. Assume that G[N_G(T)] is cyclically 3‐connected if T is a maximal K₃ in G which is also maximal in cl(G). Then G is hamiltonian. This result is a common generalization of Kaiser et al.'s theorem [J Graph Theory 48(4) (2005), 267–276] and Pfender's theorem [J Graph Theory 49(4) (2005), 262–272]. © 2011 Wiley Periodicals, Inc. J Graph Theory     

On factors of 4‐connected claw‐free graphs*

We consider the existence of several different kinds of factors in 4‐connected claw‐free graphs. This is motivated by the following two conjectures which are in fact equivalent by a recent result of the third author. Conjecture 1 (Thomassen): Every 4‐connected line graph is hamiltonian, i.e., has a connected 2‐factor. Conjecture 2 (Matthews and Sumner): Every 4‐connected claw‐free graph is hamiltonian. We first show that Conjecture 2 is true within the class of hourglass‐free graphs, i.e., graphs that do not contain an induced subgraph isomorphic to two triangles meeting in exactly one vertex. Next we show that a weaker form of Conjecture 2 is true, in which the conclusion is replaced by the conclusion that there exists a connected spanning subgraph in which each vertex has degree two or four. Finally we show that Conjectures 1 and 2 are equivalent to seemingly weaker conjectures in which the conclusion is replaced by the conclusion that there exists a spanning subgraph consisting of a bounded number of paths © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 125–136, 2001     

Hamiltonicity and forbidden subgraphs in 4‐connected graphs

Let T be the line graph of the unique tree F on 8 vertices with degree sequence (3,3,3,1,1,1,1,1), i.e., T is a chain of three triangles. We show that every 4‐connected {T, K_1,3}‐free graph has a hamiltonian cycle. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 262–272, 2005     

On Hamiltonicity of 3-Connected Claw-Free Graphs

Lai, Shao and Zhan (J Graph Theory 48:142–146, 2005) showed that every 3-connected N ₂-locally connected claw-free graph is Hamiltonian. In this paper, we generalize this result and show that every 3-connected claw-free graph G such that every locally disconnected vertex lies on some induced cycle of length at least 4 with at most 4 edges contained in some triangle of G is Hamiltonian. It is best possible in some sense.     

Line graphs of multigraphs and Hamilton‐connectedness of claw‐free graphs

We introduce a closure concept that turns a claw‐free graph into the line graph of a multigraph while preserving its (non‐)Hamilton‐connectedness. As an application, we show that every 7‐connected claw‐free graph is Hamilton‐connected, and we show that the well‐known conjecture by Matthews and Sumner (every 4‐connected claw‐free graph is hamiltonian) is equivalent with the statement that every 4‐connected claw‐free graph is Hamilton‐connected. Finally, we show a natural way to avoid the non‐uniqueness of a preimage of a line graph of a multigraph, and we prove that the closure operation is, in a sense, best possible. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:152‐173, 2011     

Degree conditions on claws and modified claws for hamiltonicity of graphs

Ore presented a degree condition involving every pair of nonadjacent vertices for a graph to be hamiltonian. Fan [New sufficient conditions for cycles in graphs, J. Combin. Theory Ser. B 37 (1984) 221-227] showed that not all the pairs of nonadjacent vertices are required, but only the pairs of vertices at the distance two suffice. Bedrossian et al. [A generalization of Fan's condition for hamiltonicity, pancyclicity, and hamiltonian connectedness, Discrete Math. 115 (1993) 39-50] improved Fan's result involving the pairs of vertices contained in an induced claw or an induced modified claw. On the other hand, Matthews and Sumner [Longest paths and cycles in K_1,3-free graphs, J. Graph Theory 9 (1985) 269-277] gave a minimum degree condition for a claw-free graph to be hamiltonian. In this paper, we give a new degree condition in an induced claw or an induced modified claw ensuring the hamiltonicity of graphs which extends both results of Bederossian et al. and Matthews and Sumner.     

Collapsible graphs and reductions of line graphs

A graph G is collapsible if for every even subset X⊆V(G), G has a subgraph Γ such that G−E(Γ) is connected and the set of odd-degree vertices of Γ is X. A graph obtained by contracting all the non-trivial collapsible subgraphs of G is called the reduction of G. In this paper, we characterize graphs of diameter two in terms of collapsible subgraphs and investigate the relationship between the line graph of the reduction and the reduction of the line graph. Our results extend former results in [H.-J. Lai, Reduced graph of diameter two, J. Graph Theory 14 (1) (1990) 77-87], and in [P.A. Catlin, Iqblunnisa, T.N. Janakiraman, N. Srinivasan, Hamilton cycles and closed trails in iterated line graphs, J. Graph Theory 14 (1990) 347-364].     

A class of symmetric graphs with 2‐arc transitive quotients

Let Γ be an X‐symmetric graph admitting an X‐invariant partition ?? on V(Γ) such that Γ_?? is connected and (X, 2)‐arc transitive. A characterization of (Γ, X, ??) was given in [S. Zhou Eur J Comb 23 (2002), 741–760] for the case where |B|>|Γ(C)∩B|=2 for an arc (B, C) of Γ_??.We con‐sider in this article the case where |B|>|Γ(C)∩B|=3, and prove that Γ can be constructed from a 2‐arc transitive graph of valency 4 or 7 unless its connected components are isomorphic to 3 K ₂, C ₆ or K _{3, 3}. As a byproduct, we prove that each connected tetravalent (X, 2)‐transitive graph is either the complete graph K ₅ or a near n‐gonal graph for some n?4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 232–245, 2010     

Connected even factors in claw-free graphs

A connected even [2,2s]-factor of a graph G is a connected factor with all vertices of degree i (i=2,4,…,2s), where s?1 is an integer. In this paper, we show that every supereulerian K_1,s-free graph (s?2) contains a connected even [2,2s-2]-factor, hereby generalizing the result that every 4-connected claw-free graph has a connected [2,4]-factor by Broersma, Kriesell and Ryjacek.     

Degree conditions restricted to induced paths for hamiltonicity of claw-heavy graphs

Broersma and Veldman proved that every 2-connected claw-free and P ₆-free graph is hamiltonian. Chen et al. extended this result by proving every 2-connected claw-heavy and P ₆-free graph is hamiltonian. On the other hand, Li et al. constructed a class of 2-connected graphs which are claw-heavy and P ₆-o-heavy but not hamiltonian. In this paper, we further give some Ore-type degree conditions restricting to induced copies of P ₆ of a 2-connected claw-heavy graph that can guarantee the graph to be hamiltonian. This improves some previous related results.     

Hamilton cycles in prisms

The prism over a graph G is the Cartesian product G □ K₂ of G with the complete graph K₂. If G is hamiltonian, then G□K₂ is also hamiltonian but the converse does not hold in general. Having a hamiltonian prism is shown to be an interesting relaxation of being hamiltonian. In this article, we examine classical problems on hamiltonicity of graphs in the context of having a hamiltonian prism. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 249–269, 2007     

Traceability of line graphs

Brualdi and Shanny [R.A. Brualdi, R.F. Shanny, Hamiltonian line graphs, J. Graph Theory 5 (1981) 307-314], Clark [L. Clark, On hamitonian line graphs, J. Graph Theory 8 (1984) 303-307] and Veldman [H.J. Veldman, On dominating and spanning circuits in graphs, Discrete Math. 124 (1994) 229-239] gave minimum degree conditions of a line graph guaranteeing the line graph to be hamiltonian. In this paper, we investigate the similar conditions guaranteeing a line graph to be traceable. In particular, we show the following result: let G be a simple graph of order n and L(G) its line graph. If n is sufficiently large and, either ; or and G is almost bridgeless, then L(G) is traceable. As a byproduct, we also show that every 2-edge-connected triangle-free simple graph with order at most 9 has a spanning trail. These results are all best possible.     

Closure, stability and iterated line graphs with a 2-factor

We consider 2-factors with a bounded number of components in the n-times iterated line graph Lⁿ(G). We first give a characterization of graph G such that Lⁿ(G) has a 2-factor containing at most k components, based on the existence of a certain type of subgraph in G. This generalizes the main result of [L. Xiong, Z. Liu, Hamiltonian iterated line graphs, Discrete Math. 256 (2002) 407-422]. We use this result to show that the minimum number of components of 2-factors in the iterated line graphs Lⁿ(G) is stable under the closure operation on a claw-free graph G. This extends results in [Z. Ryjá?ek, On a closure concept in claw-free graphs, J. Combin. Theory Ser. B 70 (1997) 217-224; Z. Ryjá?ek, A. Saito, R.H. Schelp, Closure, 2-factors and cycle coverings in claw-free graphs, J. Graph Theory 32 (1999) 109-117; L. Xiong, Z. Ryjá?ek, H.J. Broersma, On stability of the hamiltonian index under contractions and closures, J. Graph Theory 49 (2005) 104-115].     

Claw‐free 3‐connected P11‐free graphs are hamiltonian

We show that every 3‐connected claw‐free graph which contains no induced copy of P₁₁ is hamiltonian. Since there exist non‐hamiltonian 3‐connected claw‐free graphs without induced copies of P₁₂ this result is, in a way, best possible. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 111–121, 2004     

Hamilton connectivity of line graphs and claw‐free graphs

Let G be a graph and let V₀ = {ν∈ V(G): d_G(ν) = 6}. We show in this paper that: (i) if G is a 6‐connected line graph and if |V₀| ≤ 29 or G[V₀] contains at most 5 vertex disjoint K₄'s, then G is Hamilton‐connected; (ii) every 8‐connected claw‐free graph is Hamilton‐connected. Several related results known before are generalized. © 2005 Wiley Periodicals, Inc. J Graph Theory     

An excluded minor theorem for the Octahedron plus an edge

Let G be the unique 4‐connected simple graph obtained by adding an edge to the Octahedron. Every 4‐connected graph that does not contain a minor isomorphic to G is either planar or the square of an odd cycle. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 124–130, 2008