It is proven that if G is a 3-connected claw-free graph which is also Z 3 -free (where Z 3 is a triangle with a path of length 3 attached), P 6 -free (where P 6 is a path with 6 vertices) or H 1 -free (where H 1 consists of two disjoint triangles connected by an edge), then G is hamiltonian-connected. Also, examples will be described that determine a finite family of graphs L such that if a 3-connected graph being claw-free andKeywords : hamiltonian-connected, forbidden subgraph, claw-free graph. AMS Subject Classifications (1991):
Recently, Ryjfi6ek introduced an interesting new closure concept for claw-free graphs, and used it to prove that every nonhamiltonian claw-free graph is a spanning subgraph of a nonhamiltonian line graph (of a triangle-free graph). We discuss the relationship between Ryjfi~ek's closure and the Ka-closure introduced by the first author. Our main result deals with a variation on the Ka-closure. It implies a simpler proof of Ryjfi6ek's closure theorem, and yields a more general closure concept which is not restricted to claw-free graphs only. (~
A graph G on n vertices is called subpano'clic if it contains cycles of' every length k with 3 <~k ~c(G), where c(G) denotes the length of a longest cycle in G; if moreover c(G)= n, then G is called pancyclic. By a result of Flandrin et al. a claw-free graph (on at least 35 vertices) with minimum degree at least ½(n-2) is pancyclic. This degree bound is best possible. We prove that for a claw-free graph to be subpancyclic we only need the degree condition 6 > v/~n + 1 2. Again, this degree bound is best possible. It follows directly that under the same condition a hamiltonian claw-free graph is pancyclic.
Let G be a graph and let t ≥ 0 be a real number. Then, G is t‐tough if tω(G − S) ≤ |S| for all S ⊆ V(G) with ω(G − S) > 1, where ω(G − S) denotes the number of components of G − S. The toughness of G, denoted by τ(G), is the maximum value of t for which G is t‐tough [taking τ(Kn) = ∞ for all n ≥ 1]. G is minimally t‐tough if τ(G) = t and τ(H) < t for every proper spanning subgraph H of G. We discuss how the toughness of (spanning) subgraphs of G and related graphs depends on τ(G), we give some sufficient degree conditions implying that τ(G) ≥ t, and we study which subdivisions of 2‐connected graphs have minimally 2‐tough squares. © 1999 John Wiley & Sons, Inc. Networks 33: 233–238, 1999
A graph G on n vertices is called subpano'clic if it contains cycles of' every length k with 3 <~k ~c(G), where c(G) denotes the length of a longest cycle in G; if moreover c(G)= n, then G is called pancyclic. By a result of Flandrin et al. a claw-free graph (on at least 35 vertices) with minimum degree at least ½(n-2) is pancyclic. This degree bound is best possible. We prove that for a claw-free graph to be subpancyclic we only need the degree condition 6 > v/~n + 1 2. Again, this degree bound is best possible. It follows directly that under the same condition a hamiltonian claw-free graph is pancyclic.
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