Spider graphs are the intersection graphs of subtrees of subdivisions of stars. Thus, spider graphs are chordal graphs that form a common superclass of interval and split graphs. Motivated by previous results on the existence of Hamilton cycles in interval, split and chordal graphs, we show that every 3/2-tough spider graph is hamiltonian. The obtained bound is best possible since there are (3/...

The multi-path algorithm for finding hamilton cycles in a graph G is described in the book by Christofides. It is an intelligent exhaustive search for a hamilton cycle. In this paper we describe how the algorithm can be improved in two ways: (1) by detecting small separating sets M for which G–M has more than |M| components; and (2) by detecting bipartitions (X,Y), where |X|<|Y|.

We first consider the following problem. We are given a fixed perfect matching M of [n] and we add random edges one at a time until there is a Hamilton cycle containing M . We show that w.h.p. the hitting time for this event is the same as that for the first time there are no isolated vertices in the graph induced by the random edges. We then use this result for the following problem. We genera...

For 1 ≤ d ≤ ` < k, we give a new lower bound for the minimum d-degree threshold that guarantees a Hamilton `-cycle in k-uniform hypergraphs. When k ≥ 4 and d < ` = k − 1, this bound is larger than the conjectured minimum d-degree threshold for perfect matchings and thus disproves a wellknown conjecture of Rödl and Ruciński. Our (simple) construction generalizes a construction of Katona and Kier...

We describe a polynomial (O( n’.‘)) time algorithm DHAM for finding hamilton cycles in digraphs. For digraphs chosen uniformly at random from the set of digraphs with vertex set (1,2,. . . , n } and m = m(n) edges the limiting probability (as n + co) that DHAM finds a hamilton cycle equals the limiting probability that the digraph is hamiltonian. Some applications to random “ travelling salesma...

Tutte has shown that every 4-connected planar graph contains a Hamilton cycle. Grünbaum and Nash-Williams independently conjectured that the same is true for toroidal graphs. In this paper, we prove that every 4-connected toroidal graph contains a Hamilton path. Partially supported by NSF grant DMS-9970514 Partially supported by NSF grants DMS-9970527 and DMS-0245530 Partially supported by RGC ...

Given an undirected graph G = (V,E) with a weight function c ∈ R , and a positive integerK, the Kth Traveling Salesman Problem (KthTSP) is to findK Hamilton cyclesH1, H2, , ..., HK such that, for any Hamilton cycle H 6∈ {H1, H2, , ..., HK}, we have c(H) ≥ c(Hi), i = 1, 2, ...,K. This problem is NP-hard even for K fixed. We prove that KthTSP is pseudopolynomial when TSP is polynomial. 2010 Mathe...

The complexity theory of counting contrasts intriguingly with that of existence or optimization. 1. Counting versus existence The branch of theoretical computer science known as computational complexity is concerned with quantifying the computational resources required to achieve specified computational goals. Classically, the goal is often to decide the existence of a certain combinatorial str...

Select four perfect matchings of 2n vertices, independently at random. We find the asymptotic probability that each of the first and second matchings forms a Hamilton cycle with each of the third and fourth. This is generalised to embrace any fixed number of perfect matchings, where a prescribed set of pairs of matchings must each produce Hamilton cycles (with suitable restrictions on the presc...

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 set of integers m for which a graph T contains a maximal set of m edge-disjoint hamilton cycles has previously been determined whenever T is a complete graph, a complete bipartite graph, and in many cases when T is a complete m...