On Gallai's and Haj\'os' Conjectures for graphs with treewidth at most 3

TL;DR

This paper proves Gallai's and Hajós' conjectures for graphs with treewidth at most 3, extending their validity to a broader class of graphs and providing new proofs for specific subclasses.

Contribution

It verifies Gallai's and Hajós' conjectures for graphs with treewidth at most 3 and identifies the exceptional graphs that do not admit certain path decompositions.

Findings

Gallai's and Hajós' conjectures hold for graphs with treewidth ≤ 3.

Only K_3 and K_5-e are exceptions among these graphs.

New proofs are provided for graphs with maximum degree ≤ 4 and planar graphs with girth ≥ 6.

Abstract

A path (resp. cycle) decomposition of a graph $G$ is a set of edge-disjoint paths (resp. cycles) of $G$ that covers the edge set of $G$. Gallai (1966) conjectured that every graph on $n$ vertices admits a path decomposition of size at most $\lfloor (n+1)/2\rfloor$, and Haj\'os (1968) conjectured that every Eulerian graph on $n$ vertices admits a cycle decomposition of size at most $\lfloor (n-1)/2\rfloor$. Gallai's Conjecture was verified for many classes of graphs. In particular, Lov\'asz (1968) verified this conjecture for graphs with at most one vertex of even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex of odd degree. Haj\'os' Conjecture, on the other hand, was verified only for graphs with maximum degree $4$ and for planar graphs. In this paper, we verify Gallai's and Haj\'os' Conjectures for graphs with treewidth at most $3$. Moreover, we show that the only graphs with treewidth at most $3$ that do not admit a path decomposition of size at most $\lfloor n/2\rfloor$ are isomorphic to $K_3$ or $K_5-e$. Finally, we use the technique developed in this paper to present new proofs for Gallai's and Haj\'os' Conjectures for graphs with maximum degree at most $4$, and for planar graphs with girth at least $6$.