The $g$-extra edge-connectivity of balanced hypercubes

Abstract

The $g$-extra edge-connectivity is an important measure for the reliability of interconnection networks. Recently, Yang et al. [Appl. Math. Comput. 320 (2018) 464--473] determined the $3$-extra edge-connectivity of balanced hypercubes $B{H}_n$ and conjectured that the $g$-extra edge-connectivity of $B{H}_n$ is ${\lambda }_g(B{H}_n)=2(g+1)n-4g+4$ for $2\le g\le 2n-1$. In this paper, we confirm their conjecture for $n\ge 6-\frac{12}{g+1}$ and $2\le g\le 8$, and disprove their conjecture for $n\ge \frac{3e_g(B{H}_n)}{g+1}$ and $9\le g\le 2n-1$, where $e_g(B{H}_n)=\max \{|E(B{H}_n[U])|\mid U\subseteq V(B{H}_n),|U|=g+1\}$.

12 pages, 2 figures