Cycle double covers and the semi-Kotzig frame

Let H be a cubic graph admitting a 3-edge-coloring c : E(H) → Z3 such that the edges colored with 0 and μ ∈ {1, 2} induce a Hamilton circuit of H and the edges colored with 1 and 2 induce a 2-factor F . The graph H is semi-Kotzig if switching colors of edges in any even subgraph of F yields a new 3-edge-coloring ofH having the same property as c . A spanning subgraphH of a cubic graph G is called a semi-Kotzig frame if the contracted graph G/H is even and every non-circuit component of H is a subdivision of a semi-Kotzig graph. In this paper, we show that a cubic graph G has a circuit double cover if it has a semi-Kotzig frame with at most one noncircuit component. Our result generalizes some results of Goddyn [L.A. Goddyn, Cycle covers of graphs, Ph.D. Thesis, University of Waterloo, 1988], and Häggkvist and Markström [R. Häggkvist, K. Markström, Cycle double covers and spanning minors I, J. Combin. Theory Ser. B 96 (2006) 183–206]. © 2011 Elsevier Ltd. All rights reserved.