Gallai's Path Decomposition of Levi Graph
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912515368157184 |
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| author | Sahu, Akankshya Padinhatteeri, Sajith |
| author_facet | Sahu, Akankshya Padinhatteeri, Sajith |
| contents | Gallai's path decomposition conjecture states that for a connected graph $G$ on $n$ vertices, there exists a path decomposition of size $\lceil \frac{n}{2} \rceil$. The Levi graph of order one, denoted by $L_{1}(m,k)$, is a bipartite graph with vertex partition $(A,B)$, where $A$ is the collection of all $(k-1)$-element subsets of $[m]$, and $B$ is the collection of all $k$-element subsets of $[m]$. In this graph, a $(k-1)$-element subset is adjacent to a $k$-element subset if and only if it is properly contained within the $k$-element subset. The path number of a graph $G$ is the minimum size of its path decomposition. Gallai's conjecture can be seen as a conjecture on the upper bound of the path number of a connected graph. In this work, we prove the conjecture for $L_{1}(m,k)$ for all $m \ge 2 $ and $2 \le k \le m$. Moreover, we determine the path number of $L_{1}(m,2)$ for all $m$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06298 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gallai's Path Decomposition of Levi Graph Sahu, Akankshya Padinhatteeri, Sajith Combinatorics Gallai's path decomposition conjecture states that for a connected graph $G$ on $n$ vertices, there exists a path decomposition of size $\lceil \frac{n}{2} \rceil$. The Levi graph of order one, denoted by $L_{1}(m,k)$, is a bipartite graph with vertex partition $(A,B)$, where $A$ is the collection of all $(k-1)$-element subsets of $[m]$, and $B$ is the collection of all $k$-element subsets of $[m]$. In this graph, a $(k-1)$-element subset is adjacent to a $k$-element subset if and only if it is properly contained within the $k$-element subset. The path number of a graph $G$ is the minimum size of its path decomposition. Gallai's conjecture can be seen as a conjecture on the upper bound of the path number of a connected graph. In this work, we prove the conjecture for $L_{1}(m,k)$ for all $m \ge 2 $ and $2 \le k \le m$. Moreover, we determine the path number of $L_{1}(m,2)$ for all $m$. |
| title | Gallai's Path Decomposition of Levi Graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2409.06298 |