Up-Embeddability of Graphs with New Degree-Sum of Independent Vertices

Abstract:

      Let $G$ be a $k$($k\leq 3$)-edge connected simple graph with minimal degree $\geq3$, girth $g$, $r=\lfloor\frac{g-1}{2}\rfloor$. For any independent set $\{a_1,a_2,\ldots,a_{6/(4-k)}\}$ of $G$, if $$ \sum_{i=1}^{6/(4-k)}d_G(a_i)>\frac{(4-k)\nu(G)-6(g-2r-\lfloor\frac{k}{3}\rfloor)}{(4-k)(2^r-1)(g-2r)}+\frac{6}{(4-k)}(g-2r-1), $$ then $G$ is up-embeddable.