In this note, we verify two conjectures of Catlin in [J.Graph Theory \(13 (1989)465-483\)] for graphs with at most \(11\) vertices. These are used to prove the following theorem, which improves prior results in \([10]\) and \([13]\):
Let \(G\) be a 3-edge-connected simple graph with order \(n\). If \(n\) is large and if for every edge \(uv \in E(G)\), \(d(u) + d(v) \geq \frac{n}{6} – 2\), then either \(G\) has a spanning eulerian subgraph or $G$ can be contracted to the Petersen graph.