Given 0 < p < 1, we prove that a pseudo-random graph G with edge density p and sufficiently large order has the following property: Consider any red/blue-colouring of the edges of G and let r denote the proportion of edges which have colour red. Then there is a Hamilton cycle C so that the proportion of red edges of C is close to r. The analogue also holds for perfect matchings instead of Hamil...

continue to investigate the problem of the existence of a Hamilton connected cubic (m,n)-metacirculant graphs. We show that a connected cubic (m,n)-metacirculant graph G = MC(m, n, a, So, ,.., has Hamilton cycle if either a 2 == 1 (mod n) or in the case of an odd number f.L one of the numbers (a 1) or a + a 2 _ ... ajJ.-2 + ajJ.-l) relatively to n. As a corollary of results we obtain that every...

We consider the minimum cycle factor problem: given a digraph D, find the minimum number kmin(D) of vertex disjoint cycles covering all vertices of D or verify that D has no cycle factor. There is an analogous problem for paths, known as the minimum path factor problem. Both problems are NP-hard for general digraphs as they include the Hamilton cycle and path problems, respectively. In 1994 Gut...

In the Travelling Salesman Problem (TSP), we are given a complete graph Kn together with an integer weighting w on the edges of Kn, and we are asked to find a Hamilton cycle of Kn of minimum weight. Let h(w) denote the average weight of a Hamilton cycle of Kn for the weighting w. Vizing (1973) asked whether there is a polynomialtime algorithm which always finds a Hamilton cycle of weight at mos...

In the random hypergraph H = Hn,p;3 each possible triple appears independently with probability p. A loose Hamilton cycle can be described as a sequence of edges {xi, yi, xi+1} for i = 1, 2, . . . , n/2 where x1, x2, . . . , xn/2, y1, y2, . . . , yn/2 are all distinct. We prove that there exists an absolute constant K > 0 such that if p > K logn n then lim n→∞ 4|n Pr(Hn,p;3 contains a loose Ham...

The main contribution of this paper is a new approach for enumerating Hamilton cycles in bounded degree graphs – deriving thereby extremal bounds. We describe an algorithm which enumerates all Hamilton cycles of a given 3-regular n-vertex graph in time O(1.276), improving on Eppstein’s previous bound. The resulting new upper bound of O(1.276) for the maximum number of Hamilton cycles in 3-regul...

Let G (V, E) be a simple graph with vertex set V and edge set E. A generalized cycle is a subgraph such that any vertex degree is even. A simple cycle (briefly in a cycle) is a connected subgraph such that every vertex has degree 2. A basis of the cycle space is called a cycle basis of G (V, E). A cycle basis where the sum of the weights of the cycles is minimal is called a minimum cycle basis ...

A Hamilton cycle in a digraph is a cycle passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Posá ‘rotationextension’ te...