Three-regular Subgraphs of Four-regular Graphs'

The Berge–Sauer conjecture (see [2, 3]) says that any simple (no multiple edges and loops) 4-regular graph contains a 3-regular subgraph. This conjecture was proved in [4, 6]. In [1, 2] the Chevalley–Warning theorem was used to extend this result to graphs with multiple edges, which are 4-regular plus an edge. Our main result, Theorem 2.2, presents the sufficient condition for a 4-regular graph with multiple edges to have a 3-regular subgraph. It gives the new 4-regular graphs with multiple edges which have no 3-regular subgraphs, for which we know exactly the number of Euler orientations. A conjecture by Thomassen [5] says that any 4-regular connected graph with multiple edges and loops has a 3-regular subgraph, whenever the number of vertices is even. Here we reduce this conjecture to the same conjecture for graphs with only multiple edges, and one of the authors proved this conjecture [7].