# Decomposing tournaments into paths

## TL;DR

This paper investigates the minimum number of paths needed to decompose a general tournament, extending Kelly's conjecture and proving many cases of a related open conjecture.

## Contribution

It advances understanding of path decompositions in tournaments by proving numerous cases of a longstanding conjecture related to minimum path counts.

## Key findings

- Proved many cases of the conjecture for even order tournaments.
- Established lower bounds based on degree sequences.
- Extended Kelly's conjecture to path decompositions.

## Abstract

We consider a generalisation of Kelly's conjecture which is due to Alspach, Mason, and Pullman from 1976. Kelly's conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by K\"uhn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman asks for the minimum number of paths needed in a path decomposition of a general tournament $T$. There is a natural lower bound for this number in terms of the degree sequence of $T$ and it is conjectured that this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.