2-Swappability and the Edge-Reconstruction Number of Regular Graphs

TL;DR

This paper explores the relationship between edge-reconstruction and swappability in regular graphs, revealing new infinite families of graphs with higher edge-reconstruction numbers than previously conjectured.

Contribution

It establishes new theoretical links between edge-reconstruction and 2-swappability in regular graphs, leading to counterexamples to earlier conjectures.

Findings

Identified four infinite families of regular graphs with ern(G) ≥ 3

Contradicted previous upper bound conjecture for edge-reconstruction number

Connected swappability properties to edge-reconstruction in regular graphs

Abstract

The edge-reconstruction number of graph $G$, denoted $ern(G)$,is the size of the smallest multiset of edge-deleted, unlabeled subgraphs of $G$, from which the structure of $G$ can be uniquely determined. That there was some connection between the areas of edge reconstruction and swappability has been known since the swapping number of a graph was first introduced by Froncek, Rosenberg, and Hlavacek in 2013. This paper illustrates the depth of that connection by proving several bridging results between those areas; in particular, when the graphs in question are both regular and 2-swappable. These connections led to the discovery of four infinite families of $r\geq 3$ regular graphs with $ern(G) \geq 3$, contradicting the formerly conjectured upper bound.