Three-regular Subgraphs of Four-regular Graphs'

For any 4-regular graphG(possibly with multiple edges), we prove that, if the numberNof distinct Euler orientations ofGis such thatN?1 (mod 3), thenGhas a 3-regular subgraph. It gives the new 4-regular graphs with multiple edges which have no 3-regular subgraphs, for which we know the number of Euler orientations.

[1]  Noga Alon,et al.  Every 4-regular graph plus an edge contains a 3-regular subgraph , 1984, Journal of combinatorial theory. Series B (Print).

[2]  Noga Alon,et al.  Regular subgraphs of almost regular graphs , 1984, J. Comb. Theory, Ser. B.

[3]  Carsten Thomassen,et al.  Three-regular subgraphs of four-regular graphs , 1979, J. Graph Theory.

[4]  Ng Lx.  EVERY 4-REGULAR SIMPLE GRAPH CONTAINS A 3-REGULAR SUBGRAPH , 1986 .