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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.12806 |
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| _version_ | 1866915555466805248 |
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| author | Chu, Yanan Wang, Yan |
| author_facet | Chu, Yanan Wang, Yan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12806 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Path decompositions of Eulerian graphs Chu, Yanan Wang, Yan Combinatorics 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. |
| title | Path decompositions of Eulerian graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.12806 |