Upper embeddability, girth and the degree-sum of nonadjacent vertices (Q1043810)

Let \(G\) be a simple 2-edge-connedted graph of order \(v\), girth \(g\), and vertex-independence number a such that either \(\alpha\leq 2\), or \(\alpha\geq 3\) and the vertex-degree sum of any three nonadjacent vertices is at least \(v- 3g+ 7\). Then the authors show that \(G\) is upper imbeddable and that the lower bound \(v- 3g+ 7\) is best possible. They also obtain a similar result for simple 3-edge-connected graphs.