Displaying similar documents to “Pancyclism and small cycles in graphs”

A conjecture on cycle-pancyclism in tournaments

Hortensia Galeana-Sánchez, Sergio Rajsbaum (1998)

Discussiones Mathematicae Graph Theory

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Let T be a hamiltonian tournament with n vertices and γ a hamiltonian cycle of T. In previous works we introduced and studied the concept of cycle-pancyclism to capture the following question: What is the maximum intersection with γ of a cycle of length k? More precisely, for a cycle Cₖ of length k in T we denote I γ ( C ₖ ) = | A ( γ ) ∩ A ( C ₖ ) | , the number of arcs that γ and Cₖ have in common. Let f ( k , T , γ ) = m a x I γ ( C ₖ ) | C ₖ ⊂ T and f(n,k) = minf(k,T,γ)|T is a hamiltonian tournament with n vertices, and γ a hamiltonian cycle of T. In previous...

On theH-Force Number of Hamiltonian Graphs and Cycle Extendability

Erhard Hexel (2017)

Discussiones Mathematicae Graph Theory

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The H-force number h(G) of a hamiltonian graph G is the smallest cardinality of a set A ⊆ V (G) such that each cycle containing all vertices of A is hamiltonian. In this paper a lower and an upper bound of h(G) is given. Such graphs, for which h(G) assumes the lower bound are characterized by a cycle extendability property. The H-force number of hamiltonian graphs which are exactly 2-connected can be calculated by a decomposition formula.

A note on a new condition implying pancyclism

Evelyne Flandrin, Hao Li, Antoni Marczyk, Mariusz Woźniak (2001)

Discussiones Mathematicae Graph Theory

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We first show that if a 2-connected graph G of order n is such that for each two vertices u and v such that δ = d(u) and d(v) < n/2 the edge uv belongs to E(G), then G is hamiltonian. Next, by using this result, we prove that a graph G satysfying the above condition is either pancyclic or isomorphic to K n / 2 , n / 2 .

Matchings Extend to Hamiltonian Cycles in 5-Cube

Fan Wang, Weisheng Zhao (2018)

Discussiones Mathematicae Graph Theory

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Ruskey and Savage asked the following question: Does every matching in a hypercube Qn for n ≥ 2 extend to a Hamiltonian cycle of Qn? Fink confirmed that every perfect matching can be extended to a Hamiltonian cycle of Qn, thus solved Kreweras’ conjecture. Also, Fink pointed out that every matching can be extended to a Hamiltonian cycle of Qn for n ∈ {2, 3, 4}. In this paper, we prove that every matching in Q5 can be extended to a Hamiltonian cycle of Q5.

Linear forests and ordered cycles

Guantao Chen, Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson, Linda Lesniak, Florian Pfender (2004)

Discussiones Mathematicae Graph Theory

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A collection L = P ¹ ∪ P ² ∪ . . . ∪ P t (1 ≤ t ≤ k) of t disjoint paths, s of them being singletons with |V(L)| = k is called a (k,t,s)-linear forest. A graph G is (k,t,s)-ordered if for every (k,t,s)-linear forest L in G there exists a cycle C in G that contains the paths of L in the designated order as subpaths. If the cycle is also a hamiltonian cycle, then G is said to be (k,t,s)-ordered hamiltonian. We give sharp sum of degree conditions for nonadjacent vertices that imply a graph is (k,t,s)-ordered hamiltonian. ...

On Vertices Enforcing a Hamiltonian Cycle

Igor Fabrici, Erhard Hexel, Stanislav Jendrol’ (2013)

Discussiones Mathematicae Graph Theory

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A nonempty vertex set X ⊆ V (G) of a hamiltonian graph G is called an H-force set of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The H-force number h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined.