Up-embeddability and Independent Number of Simple Graphs

Abstract

Let $G$ be a $k$-edge connected simple graph with $k\le 3$, minimal degree $\delta (G)\ge 3$, and girth $g$, where $r=\lfloor \frac{g-1}{2}\rfloor$. 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.