Decompositions of complete graphs into cycles of arbitrary lengths

TL;DR

This paper characterizes when complete graphs can be decomposed into cycles of specified lengths, including cases with a perfect matching, based on parity and sum conditions.

Contribution

It provides necessary and sufficient conditions for decomposing complete graphs into cycles of arbitrary lengths, extending previous results in graph theory.

Findings

Complete graphs on odd n can be decomposed into cycles of specified lengths if sum matches edges.

Complete graphs on even n can be decomposed into cycles plus a perfect matching under certain conditions.

The paper establishes exact criteria involving parity and sum constraints for such decompositions.

Abstract

We show that the complete graph on $n$ vertices can be decomposed into $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is odd, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\cdots+m_t=\binom n2$. We also show that the complete graph on $n$ vertices can be decomposed into a perfect matching and $t$ cycles of specified lengths $m_1,\ldots,m_t$ if and only if $n$ is even, $3\leq m_i\leq n$ for $i=1,\ldots,t$, and $m_1+\ldots+m_t=\binom n2-\frac n2$.