Existence Of Hamiltonian Cycle Research Articles

The number of Hamiltonian cycles in groups of Symmetry

Groups are algebraic structures in Abstract Algebra comprised of a set of elements with a binary operation that satisfies closure, associativity, identity and invertibility. Cayley graphs serve as a visualization tool for groups, as they are capable of illustrating certain structures and properties of groups geometrically. In particular, each element in a group is assigned to a vertex in Cayley graphs. By the group action of left-multiplication, distinct elements in the generating sets can act on each element to create varied directed edges (Meier[1]). By contrast, the presence of a Hamiltonian cycles within a graph demonstrates its level of connectivity. In this research paper, utilizing directed Cayley graphs, we present a series of conjectures and theorems regarding the number and existence of Hamiltonian cycles within Dihedral groups, Symmetric groups of Platonic solids and Symmetric groups. By exploring the relationship between Abstract groups and Hamiltonian graphs, this work contributes to the broader field of research pertaining to Groups of Symmetry and Geometric Group Theory.

Some results on the existence of Hamiltonian cycles in -compositions of bipartite digraphs

Let D be a digraph on n vertices s1, …, sn and let D1, …, Dn be a family of vertex-disjoint bipartite digraphs. We think of D1, …, Dn as 2-colored digraphs with the same color set. The -composition …, is the digraph obtained from D by replacing each vertex si of D by Di and adding an arc from each vertex of Di to each vertex of Dj if and only if those vertices have different color and is an arc of D (with i, and ). Notice that this is a generalization of the usual composition D, whenever each of the digraphs D1, …, Dn are k-partite, we obtain a k-partite digraph as a -composition and this digraph is a subdigraph of the usual composition …, In our case, we obtain a bipartite digraph. Particularly, the -composition …, when is the 2- or 3-cycle and each Di is a semicomplete bipartite digraph, is also a semicomplete bipartite digraph. Gutin, Häggkvist and Manoussakis proved that a semicomplete bipartite digraph has a Hamiltonian cycle if and only if it is strong and has a cycle-factor. In this article, we prove that a digraph …, where is the n-cycle and each Di is a strong semicomplete bipartite digraph, has a Hamiltonian cycle if and only if it is strong and has a cycle-factor.

Some results on the existence of Hamiltonian cycles in the generalized sum of digraphs

Let D1,…,Dk be a family of pairwise vertex-disjoint digraphs. The generalized sum of D1,…,Dk, denoted by D1⊕⋯⊕Dk, is the set of all digraphs D which satisfies: (i) V(D)=∪i=1kV(Di), (ii) D[V(Di)]≅Di for every i∈[k] and (iii) between each pair of vertices in different summands of D there is exactly one arc. When each Di has no arcs, we have that D1⊕⋯⊕Dk is a set of k-partite digraphs. Moreover, for each tournament T on k vertices, we always have that T[D1,…,Dk], the composition of digraphs, is in D1⊕⋯⊕Dn. In this work, we give some results on the existence or the non-existence of Hamiltonian cycles and cycle-factors in such digraphs. Particularly, we prove a generalization of the characterization of Hamiltonian semicomplete bipartite digraphs due to Gutin, Häggkvist and Manoussakis.

Hamilton cycles in implicit claw-heavy graphs

We define G to be implicit claw-heavy if every induced claw of G has a pair of nonadjacent vertices such that their implicit degree sum is at least |V(G)|. In this paper, we show that an implicit claw-heavy graph G is hamiltonian if we impose certain additional conditions on G involving forbidden induced subgraphs. Our result extends a previous theorem of Chen et al. (2009) [6] on the existence of hamiltonian cycles in claw-heavy graphs.

Random Threshold Graphs

We introduce a pair of natural, equivalent models for random threshold graphs and use these models to deduce a variety of properties of random threshold graphs. Specifically, a random threshold graph $G$ is generated by choosing $n$ IID values $x_1,\ldots,x_n$ uniformly in $[0,1]$; distinct vertices $i,j$ of $G$ are adjacent exactly when $x_i + x_j \ge 1$. We examine various properties of random threshold graphs such as chromatic number, algebraic connectivity, and the existence of Hamiltonian cycles and perfect matchings.

The generalized hierarchical product of graphs

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distance, radius and diameter, simple routing algorithms and some optimal communication protocols, are inherited by the generalized hierarchical product. Besides some of these properties, in this paper we study the spectrum, the existence of Hamiltonian cycles, the chromatic number and index, and the connectivity of the generalized hierarchical product.

Trees through specified vertices

We prove a conjecture of Horak that can be thought of as an extension of classical results including Dirac’s theorem on the existence of Hamiltonian cycles. Namely, we prove for 1 ≤ k ≤ n − 2 if G is a connected graph with A ⊂ V ( G ) such that d G ( v ) ≥ k for all v ∈ A , then there exists a subtree T of G such that V ( T ) ⊃ A and d T ( v ) ≤ ⌈ n − 1 k ⌉ for all v ∈ A .

Generalized honeycomb torus is Hamiltonian

Generalized honeycomb torus is a candidate for interconnection network architectures, which includes honeycomb torus, honeycomb rectangular torus, and honeycomb parallelogramic torus as special cases. Existence of Hamiltonian cycle is a basic requirement for interconnection networks since it helps map a “token ring” parallel algorithm onto the associated network in an efficient way. Cho and Hsu [Inform. Process. Lett. 86 (4) (2003) 185–190] speculated that every generalized honeycomb torus is Hamiltonian. In this paper, we have proved this conjecture.

EMBEDDING HAMILTONIAN CYCLES, LINEAR ARRAYS AND RINGS IN A FAULTY SUPERCUBE

We consider the problem of finding Hamiltonian cycles, linear arrays and rings of a faulty supercube, if any. The proof of the existence of Hamiltonian cycles in hypercubes is easy due to the fact they are symmetric graphs. Since the supercube is asymmetric, the proof of the existence of Hamiltonian cycles is not trivial. We show that for any supercube SN, where N is the number of nodes in the supercube, there exists a Hamiltonian cycle. This implies that for any r such that 3≤r≤N, there exists a cycle of r nodes in a supercube. There are embedding algorithms proposed in this paper. The embedding algorithms show a ring with any number of nodes which can be embedded in a faulty supercube with load 1, congestion 1 and dilation 4 such that O(n2-(⌊log2 m⌋)2) faults can be tolerated, where n is the dimension of the supercube and m is the number of nodes of the ring.

Geometric tree graphs of points in convex position

Given a set P of points in the plane, the geometric tree graph of P is defined as the graph T( P) whose vertices are non-crossing spanning with straight edges trees of P, and where two trees T 1 and T 2 are adjacent if T 2 = T 1 − e + f for some edges e and f. In this paper we concentrate on the geometric tree graph of a set of n points in convex position, denoted by G n . We prove several results about G n , among them the existence of Hamiltonian cycles and the fact that they have maximum connectivity.

Hamiltonian cycles in random regular graphs

The existence of Hamiltonian cycles in random vertex-labelled regular graphs is investigated. It is proved that there exists r 0≤796 such that for r≥ r 0 almost all vertex-labelled r-regular graphs with n vertices have Hamiltonian cycles as n → ∞.