In this paper we prove that the wrapped Butterry graph WBF(d;n) of degree d and dimension n is decomposable into Hamilton cycles. This answers a conjecture of D. Barth and A. Raspaud who solved the case d = 2.

An (edge-)weighted graph is a graph in which each edge e is assigned a nonnegative real number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges, and an optimal cycle is one of maximum weight. The weighted degree w(v) of a vertex v is the sum of the weights of the edges incident with v. The following weighted analogue (and generalization) of a well-known...

We show that for any fixed α>0, cherry-quasirandom 3-graphs of positive density and sufficiently large order n with minimum vertex degree αn2 have a tight Hamilton cycle. This solves conjecture Aigner-Horev Levy.

Grinberg’s theorem is a necessary condition for the planar Hamilton graphs. In this paper, we use cycle bases and removable cycles to survey cycle structures of the Hamiltonian graphs and derive an equation of the interior faces in Grinberg’s Theorem. The result shows that Grinberg’s Theorem is suitable for the connected and simple graphs. Furthermore, by adding a new constraint of solutions to...

It is shown that the Trivalent Cayley graphs, TC,,, are near recursive. In particular, TC, is a union of four copies of i”Cn_2 with additional well placed nodes. This allows one to recursively build the Hamilton cycle in TC,.

We conjecture new sufficient conditions for a digraph to have a Hamilton cycle. In view of applications, the conjecture is of interest in the areas where unitary matrices are of importance including quantum mechanics and quantum computing.

A graph is Hamiltonian if it contains a cycle passing through every vertex. One of the cornerstone results in the theory of random graphs asserts that for edge probability p logn n , the random graph G(n, p) is asymptotically almost surely Hamiltonian. We obtain the following strengthening of this result. Given a graph G = (V,E), an incompatibility system F over G is a family F = {Fv}v∈V where ...

In this paper, we build on the work of Alspach, Chen, and Dean [2] who showed that proving the hamiltonicity of the Cayley graph of the the dihedral group Dn reduces to showing that certain cubic, connected, bipartite graphs (called honeycomb toroidal graphs) are hamilton laceable; that is, any two vertices at odd distance from each other can be joined by a hamilton path. Alspach, Chen, and Dea...

We say that a 3-uniform hypergraph has a Hamilton cycle if there is a cyclic ordering of its vertices such that every pair of consecutive vertices lies in a hyperedge which consists of three consecutive vertices. Also, let C4 denote the 3-uniform hypergraph on 4 vertices which contains 2 edges. We prove that for every ε > 0 there is an n0 such that for every n n0 the following holds: Every 3-un...

Let HPn,m,k be drawn uniformly from all m-edge, k-uniform, k-partite hypergraphs where each part of the partition is a disjoint copy of [n]. We let HP (κ) n,m,k be an edge colored version, where we color each edge randomly from one of κ colors. We show that if κ = n and m = Kn log n where K is sufficiently large then w.h.p. there is a rainbow colored perfect matching. I.e. a perfect matching in...