On decomposition thresholds for odd-length cycles and other tripartite graphs

TL;DR

This paper establishes precise minimum degree thresholds for decomposing large graphs into odd cycles of length , improving previous bounds and extending results to tripartite graphs with new bounds for other tripartite structures.

Contribution

It provides exact degree thresholds for decomposing large graphs into odd cycles and extends the analysis to tripartite graphs, improving upon previous bounds.

Findings

Threshold for -cycle decomposition is /2 + 1/(2-4) + o(1).

Decomposition thresholds approach /2 more rapidly than previously known.

Results apply to tripartite graphs, providing bounds for other tripartite structures.

Abstract

An (edge) decomposition of a graph $G$ is a set of subgraphs of $G$ whose edge sets partition the edge set of $G$. Here we show, for each odd $\ell \geq 5$, that any graph $G$ of sufficiently large order $n$ with minimum degree at least $(\frac{1}{2}+\frac{1}{2\ell-4}+o(1))n$ has a decomposition into $\ell$-cycles if and only if $\ell$ divides $|E(G)|$ and each vertex of $G$ has even degree. This threshold cannot be improved beyond $\frac{1}{2}+\frac{1}{2\ell-2}$. It was previously shown that the thresholds approach $\frac{1}{2}$ as $\ell$ becomes large, but our thresholds do so significantly more rapidly. Our methods can be applied to tripartite graphs more generally and we also obtain some bounds for decomposition thresholds of other tripartite graphs.