In [2], Brousek characterizes all triples of graphs, G1, G2, G3, with Gi = K1,3 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 K1,s, s ≥ 3 such that G1, G2, G3-free graphs of sufficiently large order contain a hamiltonian cycle. In...

In this paper we extend general grid graphs to the grid graphs consist of polygons tiling on a plane, named polygonal grid graphs. With a cycle basis satisfied polygons tiling, we study the cyclic structure of Hamilton graphs. A Hamilton cycle can be expressed as a symmetric difference of a subset of cycles in the basis. From the combinatorial relations of vertices in the subset of cycles in th...

In 1979 Hakimi, Schmeichel and Thomassen proved that in a triangulation with n vertices and no “separating triangles” – that is: no cycle of length 3 such that there are vertices inside as well as outside of the cycle – there are at least n/(log2 n) different hamiltonian cycles. We introduce a new abstract counting technique for hamiltonian cycles in general graphs. This technique is based on a...

Let ( ) , G V E = be a graph of order n. A Hamiltonian cycle of G is a cycle contains every vertex in G. Two Hamiltonian cycles 1 1 2 3 1 , , ,... , n C u u u u u = and C2 = 1 2 3 1 , , ,... , n v v v v v of G are independent if 1 1 u v = and i i u v ≠ for 2 i n ≤ ≤ . A set of Hamiltonian cycles { } 1 2 3 , , ,... k C C C C of G is mutually independent if its elements are pairwise independent. ...

Agraph of order n is said to be pancyclic if it contains cycles of all lengths from three to n. Let G be a Hamiltonian graph and let x and y be vertices of G that are consecutive on some Hamiltonian cycle in G. Hakimi and Schmeichel showed (J Combin Theory Ser B 45:99–107, 1988) that if d(x) + d(y) ≥ n then either G is pancyclic, G has cycles of all lengths except n − 1 or G is isomorphic to a ...

We use [3] for terminology and notation not defined here and consider finite simple graphs only. The first major result on the existence of hamiltonian cycles in graphs embeddable in surfaces was by H. Whitney [12] in 1931, who proved that 4-connected maximal planar graphs are hamiltonian. In 1956, W.T. Tutte [10,11] generalized Whitney’s result from maximal planar graphs to arbitrary 4-connect...

Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T , and k an even number. In this paper the following question is studied: What is the maximum intersection with γ of a directed cycle of length k contained in T [V (γ)]? It is proved that for an even k in the range n+6 2 ≤ k ≤ n−2, there exists a directed cycle Ch(k) of length h(k), h(k) ∈ {k, k − 2}...

This work focuses on analyzing the dynamics of spin I=1 nuclei evolving under a simple two pulse echo cycle and a magic echo cycle by average Hamiltonian theory. The work highlights how spectral artifacts introduced by finite pulse widths can be removed by cycling the phases of the receiver and transmitter in a well defined way. In addition, it is shown that a magic echo cycle can refocus both ...

Let T be a hamiltonian bipartite tournament with n vertices, γ a hamiltonian directed cycle of T , and k an even number. In this paper, the following question is studied: What is the maximum intersection with γ of a directed cycle of length k? It is proved that for an even k in the range 4 ≤ k ≤ n+4 2 , there exists a directed cycle Ch(k) of length h(k), h(k) ∈ {k, k−2} with |A(Ch(k))∩A(γ)| ≥ h...

Let D be a directed graph of order n. An anti-directed (hamiltonian) cycleH inD is a (hamiltonian) cycle in the graph underlyingD such that no pair of consecutive arcs in H form a directed path in D. In this paper we give sufficient conditions for the existence of anti-directed hamiltonian cycles. Specifically, we prove that a directed graph D of even order n with minimum indegree and outdegree...