A Note on $\{k,n-k\}$-Factors of Regular Graphs

TL;DR

This paper constructs specific regular graphs that lack certain factors, providing a counterexample to a previously posed problem about the existence of $\

Contribution

It introduces explicit $r$-regular graphs with no $\

Findings

Constructed $r$-regular graphs without $\

Counterexample to Akbari and Kano's problem

Clarifies limitations on $\

Abstract

Let $r$ be an odd integer, and $k$ an even integer. In this note, we present $r$-regular graphs which have no $\{k,r-k\}$-factors for all $1\le k\le {r\over2}-1$. This gives a negative answer to a problem posed by Akbari and Kano recently.