Existence of connected regular and nearly regular graphs

TL;DR

This paper proves the existence of connected regular and nearly regular graphs for given parameters, depending on the parity of the product of vertices and degree, advancing understanding of graph construction conditions.

Contribution

It establishes the existence conditions for connected regular and nearly regular graphs based on vertex and degree parity, filling a gap in graph theory.

Findings

Connected $k$-regular graphs exist when $n imes k$ is even.

Connected nearly $k$-regular graphs exist when $n imes k$ is odd.

Provides constructive existence proofs for these graphs.

Abstract

For integers $k \geq 2$ and $n \geq k+1$, we prove the following: If $n\cdot k$ is even, there is a connected $k$-regular graph on $n$ vertices. If $n\cdot k$ is odd, there is a connected nearly $k$-regular graph on $n$ vertices.