On Extension of Regular Graphs

TL;DR

This paper investigates conditions under which an r-regular graph can be extended to an (r+1)-regular graph by adding edges, providing bounds and structural considerations based on the number of vertices and subgraph presence.

Contribution

It develops new conditions and bounds for extending regular graphs, including the impact of subgraphs on the extension process.

Findings

Derived an upper bound of r depending on n for extendability

Identified conditions involving induced complete bipartite subgraphs

Analyzed the role of complete subgraphs in graph extension

Abstract

In this article, we discuss when one can extend an r-regular graph to an r + 1 regular by adding edges. Different conditions on the num- ber of vertices n and regularity r are developed. We derive an upper bound of r, depending on n, for which, every regular graph G(n, r) can be extended to an r + 1-regular graph with n vertices. Presence of induced complete bipartite subgraph and complete subgraph is dis- cussed, separately, for the extension of regularity.