Gallai's path decomposition conjecture for triangle-free planar graphs

TL;DR

This paper proves Gallai's path decomposition conjecture for triangle-free planar graphs, expanding the classes of graphs for which the conjecture is confirmed.

Contribution

It establishes the validity of Gallai's Conjecture specifically for triangle-free planar graphs, a previously unverified class.

Findings

Gallai's Conjecture holds for triangle-free planar graphs.

The result extends the classes of graphs satisfying the conjecture.

Supports the conjecture's validity in broader graph families.

Abstract

A path decomposition of a graph $G$ is a collection of edge-disjoint paths of $G$ that covers the edge set of $G$. Gallai (1968) conjectured that every connected graph on $n$ vertices admits a path decomposition of cardinality at most $\lfloor (n+1)/2\rfloor$. Gallai's Conjecture has been verified for many classes of graphs. In particular, Lov\'asz (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai's Conjecture for graphs with maximum degree at most $5$, and Botler et al. (2017) verified it for graphs with treewidth at most $3$. In this paper, we verify Gallai's Conjecture for triangle-free planar graphs.