Let Cm [over(K, -)2] stand for a cycle Cm in which every vertex is replaced by two isolated vertices and every edge by K2, 2. We prove that the complete graph K8 m k + 1 can be decomposed into graphs isomorphic to Cm [over(K, -)2] for any m ≥ 3, k > 0. Decompositions of complete graphs into certain collections of even cycles are obtained as a corollary. Also some special cases of Alspach Conjecture are solved in this article. All proofs are constructive and use both graph theory and design theory techniques.
Bibliographical noteFunding Information:
Research for this article was supported by the Ministry of Education of the Czech Republic grant No. MSM6198910027. The authors wish to express their thanks to the anonymous referees, whose comments helped in improving the quality of this paper.
- Graph decompositions
- Graph labeling