Path decompositions of tournaments

TL;DR

This paper proves a longstanding conjecture that large even-order tournaments can be decomposed into paths equal to a specific measure, and provides an asymptotically optimal result for odd-order tournaments.

Contribution

It confirms the conjecture for large tournaments and extends results to odd-order cases with asymptotic optimality.

Findings

Confirmed the conjecture for sufficiently large even-order tournaments.

Established an asymptotically optimal decomposition result for odd-order tournaments.

Advances understanding of path decompositions in tournament graphs.

Abstract

In 1976, Alspach, Mason, and Pullman conjectured that any tournament $T$ of even order can be decomposed into exactly ${\rm ex}(T)$ paths, where ${\rm ex}(T):= \frac{1}{2}\sum_{v\in V(T)}|d_T^+(v)-d_T^-(v)|$. We prove this conjecture for all sufficiently large tournaments. We also prove an asymptotically optimal result for tournaments of odd order.