A note on a new condition implying pancyclism

Evelyne Flandrin; Hao Li; Antoni Marczyk; Mariusz Woźniak

Displaying similar documents to “A note on a new condition implying pancyclism”

Factors and circuits in $K_{1, 3}$ -free graphs

Zdeněk Ryjáček (1989)

Banach Center Publications

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Extension of several sufficient conditions for Hamiltonian graphs

Ahmed Ainouche (2006)

Discussiones Mathematicae Graph Theory

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Let G be a 2-connected graph of order n. Suppose that for all 3-independent sets X in G, there exists a vertex u in X such that |N(X∖u)|+d(u) ≥ n-1. Using the concept of dual closure, we prove that 1. G is hamiltonian if and only if its 0-dual closure is either complete or the cycle C₇ 2. G is nonhamiltonian if and only if its 0-dual closure is either the graph $(K_{r} \cup K ₛ \cup K ₜ) \lor K ₂$ , 1 ≤ r ≤ s ≤ t or the graph $((n + 1) / 2) K ₁ \lor K_{(n - 1) / 2}$ . It follows that it takes a polynomial time to check the hamiltonicity or the nonhamiltonicity...

Forbidden triples implying Hamiltonicity: for all graphs

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson (2004)

Discussiones Mathematicae Graph Theory

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In [2], Brousek characterizes all triples of graphs, G₁, G₂, G₃, with $G_{i} = K_{1, 3}$ for some i = 1, 2, or 3, such that all G₁G₂G₃-free graphs contain a hamiltonian cycle. In [6], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁, G₂, G₃, none of which is a $K_{1, s}$ , s ≥ 3 such that G₁, G₂, G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In this paper, a characterization will be given of all triples G₁, G₂, G₃ with none being $K_{1, 3}$ , such that all...

Hamiltonian colorings of graphs with long cycles

Ladislav Nebeský (2003)

Mathematica Bohemica

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By a hamiltonian coloring of a connected graph $G$ of order $n \geq 1$ we mean a mapping $c$ of $V (G)$ into the set of all positive integers such that $| c (x) - c (y) | \geq n - 1 - D_{G} (x, y)$ (where $D_{G} (x, y)$ denotes the length of a longest $x - y$ path in $G$ ) for all distinct $x, y \in G$ . In this paper we study hamiltonian colorings of non-hamiltonian connected graphs with long cycles, mainly of connected graphs of order $n \geq 5$ with circumference $n - 2$ .

Potential forbidden triples implying hamiltonicity: for sufficiently large graphs

Ralph J. Faudree, Ronald J. Gould, Michael S. Jacobson (2005)

Discussiones Mathematicae Graph Theory

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In [1], Brousek characterizes all triples of connected graphs, G₁,G₂,G₃, with $G_{i} = K_{1, 3}$ for some i = 1,2, or 3, such that all G₁G₂ G₃-free graphs contain a hamiltonian cycle. In [8], Faudree, Gould, Jacobson and Lesniak consider the problem of finding triples of graphs G₁,G₂,G₃, none of which is a $K_{1, s}$ , s ≥ 3 such that G₁G₂G₃-free graphs of sufficiently large order contain a hamiltonian cycle. In [6], a characterization was given of all triples G₁,G₂,G₃ with none being $K_{1, 3}$ , such that all G₁G₂G₃-free...

Variations on a sufficient condition for Hamiltonian graphs

Ahmed Ainouche, Serge Lapiquonne (2007)

Discussiones Mathematicae Graph Theory

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Given a 2-connected graph G on n vertices, let G* be its partially square graph, obtained by adding edges uv whenever the vertices u,v have a common neighbor x satisfying the condition $N_{G} (x) \subseteq N_{G} [u] \cup N_{G} [v]$ , where $N_{G} [x] = N_{G} (x) \cup x$ . In particular, this condition is satisfied if x does not center a claw (an induced $K_{1, 3}$ ). Clearly G ⊆ G* ⊆ G², where G² is the square of G. For any independent triple X = x,y,z we define σ̅(X) = d(x) + d(y) + d(z) - |N(x) ∩ N(y) ∩ N(z)|. Flandrin et al. proved that a 2-connected graph G is hamiltonian...

Problems remaining NP-complete for sparse or dense graphs

Ingo Schiermeyer (1995)

Discussiones Mathematicae Graph Theory

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For each fixed pair α,c > 0 let INDEPENDENT SET ( $m \leq c n^{α}$ ) and INDEPENDENT SET ( $m \geq (ⁿ ₂) - c n^{α}$ ) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m \leq c n^{α}$ or $m \geq (ⁿ ₂) - c n^{α}$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ( $m \leq n + c n^{α}$ ) and HAMILTONIAN PATH ( $m \leq n + c n^{α}$ ) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m \leq n + c n^{α}$ edges. For each ϵ > 0 let HAMILTONIAN CIRCUIT (m ≥ (1 - ϵ)(ⁿ₂)) and HAMILTONIAN PATH (m ≥ (1 - ϵ)(ⁿ₂)) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH...

The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

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If $G$ is a connected graph of order $n \geq 1$ , then by a hamiltonian coloring of $G$ we mean a mapping $c$ of $V (G)$ into the set of all positive integers such that $| c (x) - c (y) | \geq n - 1 - D_{G} (x, y)$ (where $D_{G} (x, y)$ denotes the length of a longest $x - y$ path in $G$ ) for all distinct $x, y \in V (G)$ . Let $G$ be a connected graph. By the hamiltonian chromatic number of $G$ we mean $min (max (c (z); z \in V (G))),$ where the minimum is taken over all hamiltonian colorings $c$ of $G$ . The main result of this paper can be formulated as follows: Let $G$ be a connected graph of order $n \geq 3$ . Assume that there exists...

Extremal problems for forbidden pairs that imply hamiltonicity

Ralph Faudree, András Gyárfás (1999)

Discussiones Mathematicae Graph Theory

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Let C denote the claw $K_{1, 3}$ , N the net (a graph obtained from a K₃ by attaching a disjoint edge to each vertex of the K₃), W the wounded (a graph obtained from a K₃ by attaching an edge to one vertex and a disjoint path P₃ to a second vertex), and $Z_{i}$ the graph consisting of a K₃ with a path of length i attached to one vertex. For k a fixed positive integer and n a sufficiently large integer, the minimal number of edges and the smallest clique in a k-connected graph G of order n that is CY-free...

On a family of cubic graphs containing the flower snarks

Jean-Luc Fouquet, Henri Thuillier, Jean-Marie Vanherpe (2010)

Discussiones Mathematicae Graph Theory

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We consider cubic graphs formed with k ≥ 2 disjoint claws $C_{i} K_{1, 3}$ (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of $C_{i}$ are joined to the three vertices of degree 1 of $C_{i - 1}$ and joined to the three vertices of degree 1 of $C_{i + 1}$ . Denote by $t_{i}$ the vertex of degree 3 of $C_{i}$ and by T the set $t ₁, t ₂, . . ., t_{k - 1}$ . In such a way we construct three distinct graphs, namely FS(1,k), FS(2,k) and FS(3,k). The graph FS(j,k) (j ∈ 1,2,3) is the graph where the set of vertices $⋃_{i = 0}^{i = k - 1} V (C_{i}) ∖ T$ induce j cycles (note...

Pancyclism and small cycles in graphs

Ralph Faudree, Odile Favaron, Evelyne Flandrin, Hao Li (1996)

Discussiones Mathematicae Graph Theory

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We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u)+d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u)+d(v) ≥ n+z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m for each 3 ≤ m ≤ max (dC(u,v)+1, [(n+19)/13]), $d_{C} (u, v)$ being the distance of u and v on a hamiltonian cycle...