A DEGREE CONDITION FOR SPANNING EULERIAN SUBGRAPHS

Let p greater-than-or-equal-to 2 be a fixed integer. Let G be a simple and 2-edge-connected graph on n vertices, and let g be the girth of G. If d(u) + d(upsilon) greater-than-or-equal-to (2/(g - 2))((n/p) - 4 + g) holds whenever uv is-not-an-element-of E(G), and if n is sufficiently large compared to p, then either G has a spanning eulerian subgraph or G can be contracted to a graph G1 of order at most p without a spanning eulerian subgraph. Furthermore, we characterize the graphs that satisfy the conditions above such that G1 has order p and does not have any spanning eulerian subgraph.