3-Regular subgraphs and (3,1)-colorings of 4-regular pseudographs

Let G be a 4-regular pseudograph. Refer as a (3, 1)-coloring of G to a coloring of its edges by several colors such that three edges of one color and one of another meet at each vertex. The properties of (3, 1)-colorings are closely connected with the presence of 3-regular subgraphs in the graph. We prove that each connected 4-regular pseudograph containing a 3-regular subgraph admits some (3, 1)-coloring. Moreover, each 4-regular pseudograph without multiple edges (but possibly with loops) has (3, 1)-coloring, which indirectly confirms the (unproven) conjecture that every such graph contains a 3-regular subgraph. We also analyze the question of the least number of colors for a (3, 1)-coloring of a given 4-regular graph. In conclusion, we prove that the existence of a (3, 1)-coloring satisfying additional requirements (an ordered (3, 1)-coloring) is equivalent to the existence of a 3-regular subgraph. © 2014, Pleiades Publishing, Ltd.