in this paper, we investigate a problem of finding natural condition to assure the product of two graphs to be hamilton-connected. we present some sufficient and necessary conditions for $gbox h$ being hamilton-connected when $g$ is a hamilton-connected graph and $h$ is a tree or $g$ is a hamiltonian graph and $h$ is $k_2$.

For each integer m ≥ 1, consider the graph Gm whose vertex set is the set N = {0, 1, 2, . . . } of natural numbers and whose edges are the pairs xy with y = x + m or y = x − m or y = mx or y = x/m. Our aim in this note is to show that, for each m, the graph Gm contains a Hamilton path. This answers a question of Lichiardopol. For each integer m ≥ 1, consider the graph Gm whose vertex set is the...

Let D be a directed graph of order n. An anti-directed Hamilton cycle H in D is a Hamilton cycle in the graph underlying D such that no pair of consecutive arcs in H form a directed path in D. We prove that if D is a directed graph with even order n and if the indegree and the outdegree of each vertex of D is at least 23n then D contains an anti-directed Hamilton cycle. This improves a bound of...

We consider how to use the Bellman residual of the dynamic programming operator to compute suboptimality bounds for solutions to stochastic shortest path problems. Such bounds have been previously established only in the special case that “all policies are proper,” in which case the dynamic programming operator is known to be a contraction, and have been shown to be easily computable only in th...

An n-bit binary Gray code is an enumeration of all n-bit binary strings so that successive elements differ in exactly one bit position; equivalently, a hamilton path in the Hasse diagram of Bn (the partially ordered set of subsets of an n-element set, ordered by inclusion.) We construct, for each n, a hamilton path in Bn with the following additional property: edges between levels i− 1 and i of...

Let G be a graph. A Hamilton path in G is a path containing every vertex of G. The graph G is traceable if it contains a Hamilton path, while G is k-traceable if every induced subgraph of G of order k is traceable. In this paper, we study hamiltonicity of k-traceable graphs. For k ≥ 2 an integer, we define H(k) to be the largest integer such that there exists a k-traceable graph of order H(k) t...