Cycle double covers and the semi-Kotzig frame

Let H be a cubic graph admitting a 3-edge-coloring c:E(H)->Z"3 such that the edges colored with 0 and @[email protected]?{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 of H having the same property as c. A spanning subgraph H 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 non-circuit component. Our result generalizes some results of Goddyn [L.A. Goddyn, Cycle covers of graphs, Ph.D. Thesis, University of Waterloo, 1988], and Haggkvist and Markstrom [R. Haggkvist, K. Markstrom, Cycle double covers and spanning minors I, J. Combin. Theory Ser. B 96 (2006) 183-206].

[1]  C. Godsil,et al.  Cycles in graphs , 1985 .

[2]  W. T. Tutte.  On the Problem of Decomposing a Graph into n Connected Factors , 1961 .

[3]  Alon Itai,et al.  Covering a Graph by Circuits , 1978, ICALP.

[4]  Jack Edmonds,et al.  Maximum matching and a polyhedron with 0,1-vertices , 1965 .

[5]  Klas Markström,et al.  Cycle double covers and spanning minors II , 2006, Discret. Math..

[6]  Brian Alspach,et al.  Graphs with the circuit cover property , 1994 .

[7]  Martin Kochol,et al.  Five Cycle Double Covers of Some Cubic Graphs , 1995, J. Comb. Theory, Ser. B.

[8]  Roland Häggkvist,et al.  Double covers of cubic graphs with oddness 4 , 2005, J. Comb. Theory, Ser. B.

[9]  Melody Chan.  A survey of the cycle double cover conjecture , 2009 .

[10]  Bill Jackson.  On circuit covers, circuit decompositions and Euler tours of graphs , 1993 .

[11]  Roland Häggkvist,et al.  Cycle double covers of graphs with disconnected frames , 2004 .

[12]  Klas Markström,et al.  Cycle double covers and spanning minors I , 2006, J. Comb. Theory, Ser. B.

[13]  Cun-Quan Zhang,et al.  Kotzig frames and circuit double covers , 2012, Discret. Math..

[14]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[15]  Gary Chartrand,et al.  Uniformly cordial graphs , 2006, Discret. Math..

[16]  G. Szekeres,et al.  Polyhedral decompositions of cubic graphs , 1973, Bulletin of the Australian Mathematical Society.

[17]  Arthur Hoffmann-Ostenhof.  Nowhere-zero flows and structures in cubic graphs , 2011 .

[18]  C. Nash-Williams.  Edge-disjoint spanning trees of finite graphs , 1961 .

[19]  Luis A. Goddyn.  Cycle double covers of graphs with Hamilton paths , 1989, J. Comb. Theory, Ser. B.

[20]  Michael Tarsi,et al.  Semi-duality and the cycle double cover conjecture , 1986, J. Comb. Theory, Ser. B.

[21]  Cun-Quan Zhang.  Integer Flows and Cycle Covers of Graphs , 1997 .

[22]  Andreas Huck.  On cycle-double covers of graphs of small oddness , 2001, Discret. Math..