More orthogonal double covers of complete graphs by Hamiltonian paths

• Published in: Discrete mathematics Vol. 308; no. 12; pp. 2502 - 2508
• Main Authors: Hartmann, Sven, Leck, Uwe, Leck, Volker
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 28.06.2008; Elsevier
• Subjects: Applied sciences; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graph decomposition; Graph theory; Information retrieval. Graph; Mathematics; ODC; Orthogonal double cover; Sciences and techniques of general use; Self-orthogonal decomposition; Theoretical computing; Graph decomposition; Orthogonal double cover; 05C70; Self-orthogonal decomposition; 05B15; ODC; Graph path; Hamiltonian cycle; Complete graph; Hamiltonian path; Decomposition method; 05B15; 05C70; Orthogonal double cover; ODC; Graph decomposition; Self-orthogonal decomposition; Integer; Hamiltonian graph; Cyclic; Subgraph; Power
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2007.05.026

An orthogonal double cover (ODC) of the complete graph K n by a graph G is a 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 4 n and of K 16 n , 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 2 n , 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 2 n , respectively, by Hamiltonian paths for all n of the form n = 4 k 2 + 1 or n = ( k 2 + 1 ) / 2 for some integer k.