Constructions for cyclic Moebius ladder systems

J.A. Gallian has proved [J.A. Gallian, Labeling prisms and prism related graphs, Congr. Numer. 59 (1987) 89–100] that every cubic graph M 2 k obtainable from a 2 k -cycle by adding its k diameters (the so-called Moebius Ladder of order 2 k ) is graceful. Here, in the case of k even, we propose a new graceful labeling that besides being simpler than Gallian’s one is able to give, at the same time, a graceful labeling of the prism of order 2 k . Most importantly in the case of k odd, namely in the bipartite case, we prove that M 2 k also admits an α -labeling. This implies that there exists a cyclic decomposition of the complete graph K 6 k t + 1 into copies of M 2 k for every pair of positive integers k and t with k odd. e cases we are able to give such decompositions also when k is even. Apart from the case of t = 1 that is an obvious consequence of the gracefulness of M 2 k , this happens, for instance, when k ≡ 2 (mod 4) and 6 k t + 1 is a prime.