A set S of edge-disjoint hamilton cycles in a graph T is said to be maximal if the hamilton cycles in S form a subgraph of T such that T − E(S) has no hamilton cycle. The spectrum of a graph T is the set of integers m such that T contains a maximal set of m edge-disjoint hamilton cycles. This spectrum has previously been determined for all complete graphs, all complete bipartite graphs, and man...

We study a certain polytope depending on graph $G$ and parameter $\beta\in(0,1)$ which arises from embedding the Hamiltonian cycle problem in discounted Markov decision process. Eshragh \emph{et al.} conjectured lower bound proportion of feasible bases corresponding to cycles set all bases. make progress towards proof conjecture by proving results about structure In particular, we prove three m...

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε, p, k, l)-pseudorandom if for all disjoint X and Y ⊆ V (G) with |X| ≥ εpkn and |Y | ≥ εpln we have e(X,Y ) = (1± ε)p|X||Y |. We prove that for all β > 0 there is an ε > 0 such that an (ε, p, 1, 2)-pseudorandom graph on n vertices with minim...

Let H (k) n,p,κ denote a randomly colored random hypergraph, constructed on the vertex set [n] by taking each k-tuple independently with probability p, and then independently coloring it with a random color from the set [κ]. Let H be a k-uniform hypergraph of order n. An `-Hamilton cycle is a spanning subhypergraph C of H with n/(k−`) edges and such that for some cyclic ordering of the vertices...