# Edge-bipancyclicity of bubble-sort star graphs

## TL;DR

This paper proves that the n-dimensional bubble-sort star graph is edge-bipancyclic for all n≥3, meaning each edge lies on cycles of all even lengths, enhancing understanding of its cycle structure.

## Contribution

The paper establishes that the bubble-sort star graph is edge-bipancyclic for n≥3, providing new insights into its cycle properties and network robustness.

## Key findings

- Each edge lies on cycles of all even lengths from 4 to n!
- Every edge is part of at least four cycles of each even length
- The graph is bipartite and (2n-3)-regular for n≥3

## Abstract

The interconnection network considered in this paper is the bubble-sort star graph. The $n$-dimensional bubble-sort star graph $BS_n$ is a bipartite and $(2n-3)$-regular graph of order $n!$. A bipartite graph $G$ is edge-bipancyclic if each edge of $G$ lies on a cycle of all even length $l$ with $4\leq l\leq |V(G)|$. In this paper, we show that the $n$-dimensional bubble-sort star graph $BS_n$ is edge-bipancyclic for $n\ge 3$ and for each even length $l$ with $4\leq l\leq n!$, every edge of $BS_n$ lies on at least four different cycles of length $l$.