Upper Embeddable Graphs Via the Degree-Sum of Adjacent Vertices

Abstract

A semi-double graph is such a connected multi-graph that each multi-edge consists of two edges. If there is at most one loop at each vertex of a semi-double graph, then this graph is called a single-petal graph. In this paper, we obtained that if \(G\) is a connected (resp. \(2\)-edge-connected, \(3\)-edge-connected) simple graph of order \(n\), then \(G\) is upper embeddable if \(d_G(u) + d_G(v) \geq \left\lceil\frac{2n-3}{2}\right\rceil\) (resp. \(d_G(u) + d_G(v) \geq \left\lceil\frac{2n-2}{3}\right\rceil, d_G(u) + d_G(v) \geq \left\lceil\frac{2n-23}{2}\right\rceil\)) for any two adjacent vertices \(u\) and \(v\) of \(G\). In addition, by means of semi-double graph and single-petal graph, the upper embeddability of multi-graph and pseudograph are also discussed in this paper.