Hamiltonicity in Split Graphs - A Dichotomy

Journal of Graph Theory, 1998

We describe a polynomial algorithm for the Hamiltonian cycle problem for semicomplete multipartite digraphs. The existence of such an algorithm was conjectured in [16] (see also ).

Operations Research Letters, 2006

We construct an exact algorithm for the Hamiltonian cycle problem in planar graphs with worst case time complexity O(c √ n ), where c is some fixed constant that does not depend on the instance. Furthermore, we show that under the exponential time hypothesis, the time complexity cannot be improved to O(c o( √ n) ).

The Hamiltonian cycle problem is to decide whether a given graph has a Hamiltonian cycle. Bertossi and Bonuccelli (1986, Information Processing Letters, 23, 195-200) proved that the Hamiltonian Cycle Problem is NP-Complete even for undirected path graphs and left the Hamiltonian cycle problem open for directed path graphs. Narasimhan (1989, Information Processing Letters, 32, 167-170) proved that the Hamiltonian Cycle Problem is NP-Complete even for directed path graphs and left the Hamiltonian cycle problem open for rooted directed path graphs. In this paper we resolve this open problem by proving that the Hamiltonian Cycle Problem is also NP-Complete for rooted directed path graphs.

Discrete Mathematics, 2009

Let G = (X, Y ) be a bipartite graph and define σ 2 2 (G) = min{d(x) + d(y) : xy / ∈ E(G), x ∈ X, y ∈ Y }. Moon and Moser showed that if G is a bipartite graph on 2n vertices such that σ 2 2 (G) ≥ n + 1 then G is hamiltonian, sharpening a classical result of Ore [6] for bipartite graphs. Here we prove that if G is a bipartite graph on 2n vertices such that σ 2 2 (G) ≥ n + 2k − 1 then G contains k edge-disjoint hamiltonain cycles. This extends the result of Moon and Moser and a result of R. Faudree, et al.

2002

Many conditions implying the existence of a Hamiltonian cycle in a graph have been generalized to yield the existence of k edge-disjoint such cycles (see [3], [2], [5], [1] and [6]). In [7], Moon and Moser have shown that if G = (X,Y ) is a bipartite graph with |X| = |Y | = n such that for any non-adjacent pair of vertices (x, y) ∈ V (X)× V (Y ), d(x) + d(y) ≥ n + 1, then G has a Hamiltonian cycle. In this paper, we prove that if the degree condition is replaced by d(x) + d(y) ≥ n + 2k − 1, (for a given natural number k), then G contains k edgedisjoint Hamiltonian Cycles. 1 Definitions and notation Let G = (X, Y ) be a balanced bipartite graph. For two given subgraphs F and H of G, we denote by FH the subgraph of G induced by the vertices of F in H (i.e. the graph with vertex set V (F ) and edge set E(F ) ∩ E(H). For any given subgraphs A and B of G, let N(A) be the set of vertices of G that are adjacent to at least one vertex of A, let E(A,B) be the set of edges that have an end ve...

Information Processing Letters, 2007

A Hamiltonian cycle is a closed path through all the vertices of a graph. Since discovering whether a graph has a Hamiltonian path or a Hamiltonian cycle are both NP-complete problems, researchers concentrated on formulating sufficient conditions that ensure Hamiltonicity of a graph. A recent paper [Rahman M. Sohel and Kaykobad M., "On Hamiltonian Cycles and Hamiltonian Paths", Information Processing Letters 94 , 37-51] presents distance based sufficient conditions for the existence of a Hamiltonian path. In this paper we establish that the same condition forces Hamiltonian cycle to be present excepting for the case where end points of a Hamiltonian path is at a distance greater than 2.

For integer k ≥ 1, a (p, q)-graph G = (V, E) is said to admit an AL(k)-traversal if there exist a sequence of vertices (v1, v2, . . . , vp) such that for each i = 1, 2, . . . , p − 1, the distance between vi and vi+1 is k. We call a graph k-step Hamiltonian (or admits a k-step Hamiltonian tour) if it admits an AL(k)-traversal and d(v1, vp) = k. In this paper we consider k-step Hamiltonicity of bipartite and tripartite graphs. As an application, we found that a 2-step Hamiltonian tour of a graph could sometimes induce a super-edge-magic labeling of the graph.

Journal of Combinatorial Theory, Series B, 1988

SpringerPlus, 2016

The Hamiltonian cycle (HC) problem has many applications such as time scheduling, the choice of travel routes and network topology . Therefore, resolving the HC is an important problem in graph theory and computer science as well . It is known to be in the class of NP-complete problems and consequently, determining if a graph is Hamiltonian, using the current algorithms, if it has a high time complexity. The difficulty of finding HC increases exponentially with the problem size. Also, there is an algorithm for solving the HC problem with polynomial expected running time . A Hamiltonian path is a path in an undirected graph that visits each vertex exactly once. A Hamiltonian cycle is the cycle that visits each vertex once. A Hamiltonian graph is a graph that has a Hamiltonian cycle . Due to their similarities, the problem of an HC is usually compared with Euler's problem, but solving them is very different. There exists a very elegant, necessary and sufficient condition for a graph to have Euler Cycles. Also, literature presents many solutions that generate efficient algorithms for finding Euler Cycles. Unfortunately, there is a lack of solutions for the Hamilton problem. In addition to the author' knowledge, there is no research indicating a necessary and sufficient condition for a graph to have a HC. Some of the theorems provide finding sufficient conditions for a graph to be Hamiltonian

International Journal of Computer Applications (ISSN: 0975 – 8887), 2015

A graph G (V, E) is said to be Hamiltonian if it contains a spanning cycle. The spanning cycle is called a Hamiltonian cycle of G and G is said to be a Hamiltonian graph. A Hamiltonian path is a path that contains all the vertices in V (G) but does not return to the vertex in which it began. In this paper, we study Hamiltonicity of 3-connected, 3-regular planar bipartite graph G with partite sets V=M  N. We shall prove that G has a Hamiltonian cycle if G is balanced with M = N. For that we present an algorithm for a bipartite graph KM,N where M>3, N>3 and M,N both are even to possess a Hamiltonian cycle. In particular, we also prove a theorem for S proper subset (M or N) of V the number of components W (G-S) = S implies the graph has a Hamiltonian path.