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On hyper-Zagreb index conditions for hamiltonicity of graphs

Yong Lu, Qiannan Zhou (2022)

Czechoslovak Mathematical Journal

During the last decade, several research groups have published results on sufficient conditions for the hamiltonicity of graphs by using some topological indices. We mainly study hyper-Zagreb index and some hamiltonian properties. We give some sufficient conditions for graphs to be traceable, hamiltonian or Hamilton-connected in terms of their hyper-Zagreb indices. In addition, we also use the hyper-Zagreb index of the complement of a graph to present a sufficient condition for it to be Hamilton-connected....

On k -ordered bipartite graphs.

Faudree, Jill R., Gould, Ronald J., Pfender, Florian, Wolf, Allison (2003)

The Electronic Journal of Combinatorics [electronic only]

On k-Path Pancyclic Graphs

Zhenming Bi, Ping Zhang (2015)

Discussiones Mathematicae Graph Theory

For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k + 1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G...

On L-ideal-based L-zero-divisor graphs

S. Ebrahimi Atani, M. Shajari Kohan (2011)

Discussiones Mathematicae - General Algebra and Applications

In a manner analogous to a commutative ring, the L-ideal-based L-zero-divisor graph of a commutative ring R can be defined as the undirected graph Γ(μ) for some L-ideal μ of R. The basic properties and possible structures of the graph Γ(μ) are studied.

On the Boolean function graph of a graph and on its complement

T. N. Janakiraman, S. Muthammai, M. Bhanumathi (2005)

Mathematica Bohemica

For any graph G , let V ( G ) and E ( G ) denote the vertex set and the edge set of G respectively. The Boolean function graph B ( G , L ( G ) , N I N C ) of G is a graph with vertex set V ( G ) ∪ E ( G ) and two vertices in B ( G , L ( G ) , N I N C ) are adjacent if and only if they correspond to two adjacent vertices of G , two adjacent edges of G or to a vertex and an edge not incident to it in G . For brevity, this graph is denoted by B 1 ( G ) . In this paper, structural properties of B 1 ( G ) and its complement including traversability and eccentricity properties are studied. In addition,...

On the parallel complexity of the alternating Hamiltonian cycle problem

E. Bampis, Y. Manoussakis, I. Milis (2010)

RAIRO - Operations Research

Given a graph with colored edges, a Hamiltonian cycle is called alternating if its successive edges differ in color. The problem of finding such a cycle, even for 2-edge-colored graphs, is trivially NP-complete, while it is known to be polynomial for 2-edge-colored complete graphs. In this paper we study the parallel complexity of finding such a cycle, if any, in 2-edge-colored complete graphs. We give a new characterization for such a graph admitting an alternating Hamiltonian cycle which allows...

On the stability for pancyclicity

Ingo Schiermeyer (2001)

Discussiones Mathematicae Graph Theory

A property P defined on all graphs of order n is said to be k-stable if for any graph of order n that does not satisfy P, the fact that uv is not an edge of G and that G + uv satisfies P implies d G ( u ) + d G ( v ) < k . Every property is (2n-3)-stable and every k-stable property is (k+1)-stable. We denote by s(P) the smallest integer k such that P is k-stable and call it the stability of P. This number usually depends on n and is at most 2n-3. A graph of order n is said to be pancyclic if it contains cycles of all lengths...

On theH-Force Number of Hamiltonian Graphs and Cycle Extendability

Erhard Hexel (2017)

Discussiones Mathematicae Graph Theory

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.

On traceability and 2-factors in claw-free graphs

Dalibor Fronček, Zdeněk Ryjáček, Zdzisław Skupień (2004)

Discussiones Mathematicae Graph Theory

If G is a claw-free graph of sufficiently large order n, satisfying a degree condition σₖ > n + k² - 4k + 7 (where k is an arbitrary constant), then G has a 2-factor with at most k - 1 components. As a second main result, we present classes of graphs ₁,...,₈ such that every sufficiently large connected claw-free graph satisfying degree condition σ₆(k) > n + 19 (or, as a corollary, δ(G) > (n+19)/6) either belongs to ⋃ ⁸ i = 1 i or is traceable.

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