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A graph G is called Hamilton-connected if for every pair of distinct vertices {u, v} of G there exists a Hamilton path in G that connects u and v. A graph G is said to be t-tough if t·ω(G − X) ≤ |X| for all X ⊆ V (G) with ω(G − X) > 1. The toughness of G, denoted τ (G), is the maximum value of t such that G is t-tough (taking τ (Kn) = ∞ for all n ≥ 1). It is known that a Hamilton-connected graph G has toughness τ (G) > 1, but that the reverse statement does not hold in general. In this paper, we investigate all possible forbidden subgraphs H such that every H-free graph G with τ (G) > 1 is Hamilton-connected. We find that the results are completely analogous to the Hamiltonian case: every graph H such that any 1-tough H-free graph is Hamiltonian also ensures that every H-free graph with toughness larger than one is Hamilton-connected. And similarly, there is no other forbidden subgraph having this property, except possibly for the graph K1 ∪ P4 itself. We leave this as an open case.

Let G be a 4-connected graph. We call an edge e of G removable if the following sequence of operations results in a 4-connected graph: delete e from G; if there are vertices with degree 3 in G− e, then for each (of the at most two) such vertex x, delete x from G − e and turn the three neighbors of x into a clique by adding any missing edges (avoiding multiple edges). In this paper, we continue the study on the distribution of removable edges in a 4-connected graph G, in particular outside a cycle of G or in a spanning tree or on a Hamilton cycle of G. We give examples to show that our results are in some sense best possible.

We introduce a new class of graphs which we call P3-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P3-dominated graph. We prove that G is hamiltonian if α( G2) ≤ κ( G), with two exceptions: K2,3 and K1,1,3. We also prove that G is hamiltonian, if G is 3-connected and | V( G)| ≤ 5 δ( G) − 5. These results extend known results on (quasi-)claw-free graphs. [ABSTRACT FROM AUTHOR]

A graph G is said to be 1-tough if for every vertex cut S of G, the number of components of G − S does not exceed |S|. Being 1-tough is an obvious necessary condition for a graph to be hamiltonian, but it is not sufficient in general. We study the problem of characterizing all graphs H such that every 1-tough H-free graph is hamiltonian. We almost obtain a complete solution to this problem, leaving H = K1 ∪ P4 as the only open case.