More orthogonal double covers of complete graphs by Hamiltonian paths

An orthogonal double cover (ODC) of the complete graph K-n by a graph G is n collection G of n spanning subgraphs of K-n, all isomorphic to G, such that any two members of G share exactly one edge and every edge of K-n is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of length n - 1. It is known that the existence of an ODC of K-n by a Hamiltonian path implies the existence of ODCs of K-4n, and of (K16n), respectively, by almost-Hamiltonian cycles. Horton and Nonay introduced two-colorable ODCs and showed: If there are an ODC of K-n by a Hamiltonian path for some n >= 3 and a two-colorable ODC of K-q by a Hamiltonian path for some prime power q >= 5, then there is an ODC of K-qn by a Hamiltonian path. In [U. Leck, A class of 2-colorable orthogonal double covers of complete graphs by hamiltonian paths, Graphs Combin. 18 (2002) 155-167], two-colorable ODCs of K-n and K-2n, respectively, by Hamiltonian paths were constructed for all odd square numbers n >= 9. Here we continue this work and construct cyclic two-colorable ODCs of K-n and K-2n, respectively, by Hamiltonian paths for all n of the form n = 4k(2) + 1 or n = (k(2) + 1)/2 for some integer k. (C) 2007 Elsevier B.V. All rights reserved.