Gallai's Path Decomposition for 2-degenerate Graphs
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914676408844288 |
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| author | Anto, Nevil Basavaraju, Manu |
| author_facet | Anto, Nevil Basavaraju, Manu |
| contents | Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_07159 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Gallai's Path Decomposition for 2-degenerate Graphs Anto, Nevil Basavaraju, Manu Combinatorics Discrete Mathematics 05C38, 05C70 G.2.2 Gallai's path decomposition conjecture states that if $G$ is a connected graph on $n$ vertices, then the edges of $G$ can be decomposed into at most $\lceil \frac{n }{2} \rceil$ paths. A graph is said to be an odd semi-clique if it can be obtained from a clique on $2k+1$ vertices by deleting at most $k-1$ edges. Bonamy and Perrett asked if the edges of every connected graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n}{2} \rfloor$ paths unless $G$ is an odd semi-clique. A graph $G$ is said to be 2-degenerate if every subgraph of $G$ has a vertex of degree at most $2$. In this paper, we prove that the edges of any connected 2-degenerate graph $G$ on $n$ vertices can be decomposed into at most $\lfloor \frac{n }{2} \rfloor$ paths unless $G$ is a triangle. |
| title | Gallai's Path Decomposition for 2-degenerate Graphs |
| topic | Combinatorics Discrete Mathematics 05C38, 05C70 G.2.2 |
| url | https://arxiv.org/abs/2211.07159 |