Approximate path decompositions of regular graphs

TL;DR

This paper demonstrates that the edges of any d-regular graph can be nearly decomposed into paths of length about d, and confirms a conjecture that most vertices can be partitioned into paths, advancing understanding of graph decompositions.

Contribution

It introduces an approximate path decomposition method for regular graphs and confirms a longstanding conjecture about vertex partitioning into paths.

Findings

Edges of d-regular graphs can be almost decomposed into paths of length roughly d.

Most vertices of a d-regular graph can be partitioned into about n/(d+1) paths.

Provides an approximate solution to a problem posed in 1957.

Abstract

We show that the edges of any $d$-regular graph can be almost decomposed into paths of length roughly $d$, giving an approximate solution to a problem of Kotzig from 1957. Along the way, we show that almost all of the vertices of a $d$-regular graph can be partitioned into $n/(d+1)$ paths, asymptotically confirming a conjecture of Magnant and Martin from 2009.