A Note on the Neighbor Condition for Up-Embedidability of Graphs

Abstract

Different neighbor conditions are considered in $[3,4,9]$ for a graph up-embeddable. In this paper, we consider the neighbor conditions of all the pairs of vertices with diameter $2$ and obtain the following new result: if $|N_G(u)\cap N_G(v)|\ge 2$ for any two vertices $u,v\in D$ where $D=\{(u,v)|d_G(u,v)=2,u,v\in V(G)\}$, then $G$ is up-embeddable.