Up-embeddability and Independent Number of Simple Graphs

Abstract

Let \(G\) be a \(k\)-edge connected simple graph with \(k \leq 3\), minimal degree \(\delta(G) \geq 3\), and girth \(g\), where \(r = \left\lfloor \frac{g-1}{2} \right\rfloor\). If the independence number \(\alpha(G)\) of \(G\) satisfies

\[\alpha(G) < \frac{6{(\delta-1)}^{\lfloor\frac{g}{2}\rfloor}-6}{(4-k)(\delta-2)} – \frac{6(g-2r-1)}{4-k} \] then \(G\) is up-embeddable.