Displaying similar documents to “Extension of several sufficient conditions for Hamiltonian graphs”

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 ) N G [ u ] N G [ v ] , where N G [ x ] = N G ( x ) 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...

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 .

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 , t , . . . , 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...

Intersection graph of gamma sets in the total graph

T. Tamizh Chelvam, T. Asir (2012)

Discussiones Mathematicae Graph Theory

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In this paper, we consider the intersection graph I Γ ( ) of gamma sets in the total graph on ℤₙ. We characterize the values of n for which I Γ ( ) is complete, bipartite, cycle, chordal and planar. Further, we prove that I Γ ( ) is an Eulerian, Hamiltonian and as well as a pancyclic graph. Also we obtain the value of the independent number, the clique number, the chromatic number, the connectivity and some domination parameters of I Γ ( ) .

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...

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 c n α ) and INDEPENDENT SET ( m ( ) - c n α ) be the problem INDEPENDENT SET restricted to graphs on n vertices with m c n α or m ( ) - c n α edges, respectively. Analogously, HAMILTONIAN CIRCUIT ( m n + c n α ) and HAMILTONIAN PATH ( m n + c n α ) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with m 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...

An upper bound of a generalized upper Hamiltonian number of a graph

Martin Dzúrik (2021)

Archivum Mathematicum

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In this article we study graphs with ordering of vertices, we define a generalization called a pseudoordering, and for a graph H we define the H -Hamiltonian number of a graph G . We will show that this concept is a generalization of both the Hamiltonian number and the traceable number. We will prove equivalent characteristics of an isomorphism of graphs G and H using H -Hamiltonian number of G . Furthermore, we will show that for a fixed number of vertices, each path has a maximal upper...

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 1 we mean a mapping c of V ( G ) into the set of all positive integers such that | c ( x ) - c ( y ) | 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 G . In this paper we study hamiltonian colorings of non-hamiltonian connected graphs with long cycles, mainly of connected graphs of order n 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...

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...

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 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 ) | 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 V ( G ) . Let G be a connected graph. By the hamiltonian chromatic number of G we mean min ( max ( c ( z ) ; z 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 3 . Assume that there exists...