Path decompositions of Eulerian graphs

Yanan Chu; Yan Wang

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arXiv · 2510.12806

Path decompositions of Eulerian graphs

Yanan Chu·Yan Wang

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Abstract

Gallai's conjecture asserts that every connected graph on $n$ vertices can be decomposed into $\frac{n+1}{2}$ paths. For general graphs (possibly disconnected), it was proved that every graph on $n$ vertices can be decomposed into $\frac{2n}{3}$ paths. This is also best possible (consider the graphs consisting of vertex-disjoint triangles). Lovász showed that every $n$-vertex graph with at most one vertex of even degree can be decomposed into $\frac{n}{2}$ paths. However, Gallai's conjecture is difficult for graphs with many vertices of even degrees. Favaron and Kouider verified Gallai's conjecture for all Eulerian graphs with maximum degree at most $4$. In this paper, we show if $G$ is an Eulerian graph on $n \ge 4$ vertices and the distance between any two triangles in $G$ is at least $3$, then $G$ can be decomposed into at most $\frac{3n}{5}$ paths.

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BibTeX RIS

Yanan Chu, Yan Wang. 2025-10-04. Path decompositions of Eulerian graphs. https://arxiv.org/abs/2510.12806

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