On path decompositions of 2k-regular graphs

TL;DR

This paper proves Gallai's conjecture for a specific class of 2k-regular graphs with certain girth and perfect matching properties, showing their edges can be partitioned into a minimal number of paths with controlled lengths.

Contribution

It establishes the validity of Gallai's conjecture for the class of 2k-regular graphs with girth at least 2k-2 and disjoint perfect matchings, including explicit path length partitions.

Findings

Gallai's conjecture holds for the class G_k of 2k-regular graphs with girth 2k-2.

Edge set of such graphs can be partitioned into n/2 paths of lengths 2k-1, 2k, 2k+1.

Confirmed the existence of path decompositions with specific length constraints.

Abstract

Tibor Gallai conjectured that the edge set of every connected graph $G$ on $n$ vertices can be partitioned into $\lceil n/2\rceil$ paths. Let $\mathcal{G}_{k}$ be the class of all $2k$-regular graphs of girth at least $2k-2$ that admit a pair of disjoint perfect matchings. In this work, we show that Gallai's conjecture holds in $\mathcal{G}_{k}$, for every $k \geq 3$. Further, we prove that for every graph $G$ in $\mathcal{G}_{k}$ on $n$ vertices, there exists a partition of its edge set into $n/2$ paths of lengths in $\{2k-1,2k,2k+1\}$.