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

TL;DR

This paper investigates the $g$-extra edge-connectivity of balanced hypercubes, confirming a conjecture for certain parameters and disproving it for others, thereby advancing understanding of network reliability measures.

Contribution

The paper proves the conjectured formula for $g$-extra edge-connectivity in specific cases and provides counterexamples in others, refining the theoretical understanding of balanced hypercube connectivity.

Findings

Confirmed the conjecture for $n extgreater 6-rac{12}{g+1}$ and $2 extless g extless 9$.

Disproved the conjecture for $n extgreater rac{3e_g(BH_n)}{g+1}$ and $9 extless g extless 2n-1$.

Enhanced the theoretical framework for analyzing network reliability 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 $BH_n$ and conjectured that the $g$-extra edge-connectivity of $BH_n$ is $\lambda_g(BH_n)=2(g+1)n-4g+4$ for $2\leq g\leq 2n-1$. In this paper, we confirm their conjecture for $n\geq 6-\dfrac{12}{g+1}$ and $2\leq g\leq 8$, and disprove their conjecture for $n\geq \dfrac{3e_g(BH_n)}{g+1}$ and $9\leq g\leq 2n-1$, where $e_g(BH_n)=\max\{|E(BH_n[U])|\mid U\subseteq V(BH_n), |U|=g+1\}$.