Sufficient Condition For Hamiltonicity Research Articles

On dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs

We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices.

Sufficient Conditions for Hamiltonicity of Graphs with Respect to Wiener Index, Hyper-Wiener Index, and Harary Index

In this paper, with respect to the Wiener index, hyper-Wiener index, and Harary index, it gives some sufficient conditions for some graphs to be traceable, Hamiltonian, Hamilton-connected, or traceable for every vertex. Firstly, we discuss balanced bipartite graphs with δG≥t, where δG is the minimum degree of G, and gain some sufficient conditions for the graphs to be traceable or Hamiltonian, respectively. Secondly, we discuss nearly balanced bipartite graphs with δG≥t and present some sufficient conditions for the graphs to be traceable. Thirdly, we discuss graphs with δG≥t and obtain some conditions for the graphs to be traceable or Hamiltonian, respectively. Finally, we discuss t-connected graphs and provide some conditions for the graphs to be Hamilton-connected or traceable for every vertex, respectively.

Leaf number and Hamiltonian $$C_4$$ C 4 -free graphs

Let G be a simple 2-connected, $$C_4$$ -free graph with minimum degree $$\delta (G)\ge 4$$ and leaf number L(G) such that $$\delta (G)\ge \displaystyle \frac{1}{2}L(G)$$ . We show that G is Hamiltonian. In addition, we provide family of graphs to show that the results are best possible for aforementioned class of graphs. The results, apart from supporting the conjecture (Graffiti.pc 190) of the computer program Graffiti.pc, instructed by DeLaVi $${\tilde{n}}$$ a, provide a new sufficient condition for Hamiltonicity in $$C_4$$ -free graphs.

Sufficient Conditions for Hamiltonicity in Communication Networks through Graph Modeling: New Research Challenges

Sufficient Conditions for Hamiltonicity in Communication Networks through Graph Modeling: New Research Challenges

Extremal and Degree Conditions for Path Extendability in Digraphs

In the study of cycles and paths, the meta-conjecture of Bondy that sufficient conditions for Hamiltonicity often imply pancyclicity has motivated research on the existence of cycles and paths of many lengths. Hendry further introduced the stronger concepts of cycle extendability and path extendability, which require that every cycle or path can be extended to another one with one additional vertex. These concepts have been studied extensively, but there exist few results on path extendability in digraphs, as far as we know. In this paper, we make the first attempt in this direction. We establish a number of extremal and degree conditions for path extendability in general digraphs. Moreover, we prove that every path of length at least two in a regular tournament is extendable, with some exceptions. One of our proof approaches is a new contraction operation to transform nonextendable paths into nonextendable cycles.

Sufficient conditions for Hamiltonicity in multiswapped networks

OTIS networks are interconnection networks amenable to deployment as hybrid networks containing both electronic and optical links. Deficiencies as regards symmetry led to the subsequent formulation of biswapped networks which were later generalized to multiswapped networks so as to still enable optoelectronic implementation (as it happens, multiswapped networks also generalize previously studied hierarchical crossed cubes). Multiswapped networks of the form Msw(H;G) are known to possess good (graph-theoretic) properties as regards their use as (optoelectronic) interconnection networks (in distributed-memory multiprocessors) and in relation to those of the component networks G and H. Combinatorially, they provide a hierarchical mechanism to define new networks from existing networks (so that the properties of the new network can be controlled in terms of the constituent networks). In this paper, we prove that if G and H are Hamiltonian networks then the multiswapped network Msw(H;G) is also Hamiltonian. At the core of our proof is finding specially designed Hamiltonian cycles in 2-dimensional and heavily pruned 3-dimensional tori, irrespective of the actual networks G and H we happen to be working with. This lends credence to the role of tori as fundamental networks within the study of interconnection networks.

Cyclic properties of the topological graphs of hexagonal grids

Under study are the cyclic properties of the topological graphs of hexagonal grids. Some sufficient condition for Hamiltonicity of these graphs is obtained. We find the minimal 2-connected non-Hamiltonian topological graph of a hexagonal grid. An upper bound is established for the shortness coefficient of the class of graphs under study.

A sufficient condition for Hamiltonicity in locally finite graphs

Using topological circles in the Freudenthal compactification of a graph as infinite cycles, we extend to locally finite graphs a result of Oberly and Sumner on the Hamiltonicity of finite graphs. This answers a question of Stein, and gives a sufficient condition for Hamiltonicity in locally finite graphs.

On spanning cycles, paths and trees

Let G be a simple connected graph with minimum degree δ. Then G is Hamiltonian if it contains a spanning cycle and traceable if it contains a spanning path. The leaf number L(G) of G is defined as the maximum number of end vertices contained in a spanning tree of G. We prove a sufficient condition, depending on L(G) and δ, for G to be Hamiltonian or traceable. Our results, apart from providing a new sufficient condition for Hamiltonicity, settle completely a conjecture of the computer program, Graffiti.pc, instructed by DeLaViña.

Normal eulerian clique-covering and hamiltonicity

A connected graph is hamiltonian if it contains a cycle which goes through all vertices exactly once. Determining if a graph is hamiltonian is known as an NP-complete problem and no satisfactory characterization for these graphs has been found. Since the seminal work of Dirac in 1952 many sufficient conditions were found. In 1974, Goodman and Hedetniemi gave such a condition based on the existence of a clique-covering of the graph. This condition was recently generalized using the notion of eulerian clique-covering. In addition, an algorithm able to find a normal eulerian clique-covering for a large class of graphs was also introduced. A normal clique-covering has additional properties, making the search for such a covering easier than in the general case. In this article, we prove several properties of normal clique-coverings. In particular we prove that there exists an eulerian clique-covering of a graph if and only if there exists a normal one. Using this result, the search for an eulerian clique-covering as a sufficient condition for hamiltonicity can be reduced to the normal case.

On a generalization of Chvátal's condition giving new hamiltonian degree sequences

By imposing a structural criterion on a graph, we generalize the well-known Chvátal's sufficient condition for hamiltonicity (J. Combin. Theory Ser. B 12 (1972) 163–168). Using this result, we describe a new class of hamiltonian degree sequences which contains the sequences given by Chvátal's condition, as well as a class of degree sequences described by Fan and Liu (J. Systems Sci. Math. Sci. 4 (1) (1984) 27–32).

Hamiltonicity of Amalgams

An amalgam is obtained from two Halin graphs having skirting cycles of the same length. We are interested in hamiltonicity of amalgams constructed from two identical Halin graphs without any shift along the skirting cycle. We establish hamiltonicity of amalagams constructed from cubic Halin graphs. We give a sufficient condition for hamiltonicity of non-cubic amalgams and characterize infinite classes of non-Hamiltonian amalgams. We also characterize hamiltonicity of amalgams constructed by shifting the component Halin graphs by one and of general amalgams of higher degree.

On a Closure Concept in Claw-Free Graphs

IfGis a claw-free graph, then there is a graphcl(G) such that (i)Gis a spanning subgraph ofcl(G), (ii)cl(G) is a line graph of a triangle-free graph, and (iii)the length of a longest cycle inGand incl(G) is the same. A sufficient condition for hamiltonicity in claw-free graphs, the equivalence of some conjectures on hamiltonicity in 2-tough graphs and the hamiltonicity of 7-connected claw-free graphs are obtained as corollaries.