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Small cycle double covers of products I: Lexicographic product with paths and cycles

✍ Scribed by R. J. Nowakowski; K. Seyffarth

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
243 KB
Volume
57
Category
Article
ISSN
0364-9024

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✦ Synopsis

Abstract

Bondy 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 □ P~2~ and G □ C~2k~ have SCDCs. If G has an SCDC then so does G □ P~k~, k > 2 and G □ C~2k + 1~. We use these Cartesian results to show that P~2j~[G] (j ≥ 1) and C~k~[G] (k ≠ 3, 5, 7) have SCDCs. Also, if G has an SCDC then so does P~2j + 1~[G] (j ≥ 4). The results for the lexicographic product are harder and, in addition to the Cartesian results, require certain decompositions of K~n,n~ into perfect matchings. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 99–123, 2008

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