On the path partition number of 6-regular graphs

TL;DR

This paper proves the path partition conjecture for 6-regular graphs, establishing that such graphs can be partitioned into at most n/7 paths, extending previous results for lower degrees.

Contribution

The paper confirms the path partition conjecture for 6-regular graphs, a case previously unresolved, thus completing the proof for all degrees less than 6.

Findings

Confirmed the conjecture for 6-regular graphs

Established a bound of n/7 paths for 6-regular graphs

Extended the proof of the conjecture to the case d=6

Abstract

A path partition (also referred to as a linear forest) of a graph $G$ is a set of vertex-disjoint paths which together contain all the vertices of $G$. An isolated vertex is considered to be a path in this case. The path partition conjecture states that every $n$-vertices $d$-regular graph has a path partition with at most $\frac{n}{d+1}$ paths. The conjecture has been proved for all $d<6$. We prove the conjecture for $d=6$.