Abstract
AbstractBondy conjectured that every simple bridgeless graph has a small cycle double cover (SCDC). We show that this is the case for the lexicographic products of certain graphs and along the way for the Cartesian product as well. Specifically, if G does not have an isolated vertex then G □ P2 and G □ C2k have SCDCs. If G has an SCDC then so does G □ Pk, k > 2 and G □ C2k + 1. We use these Cartesian results to show that P2j[G] (j ≥ 1) and Ck[G] (k ≠ 3, 5, 7) have SCDCs. Also, if G has an SCDC then so does P2j + 1[G] (j ≥ 4). The results for the lexicographic product are harder and, in addition to the Cartesian results, require certain decompositions of Kn,n into perfect matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 99–123, 2008
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