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arXiv · 2310.11704

Gallai's path decomposition conjecture for cartesian product of graphs (\uppercase\expandafter{\romannumeral 2})

Xiaohong Chen·Baoyindureng Wu
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Abstract

Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is a collection of edge-disjoint paths that covers all the edges of $G$. Let $p(G)$ denote the minimum number of paths needed in a path decomposition of $G$. Gallai conjectured that if $G$ is connected, then $p(G)\leq \lceil\frac{n}{2}\rceil$. In this paper, we prove that Gallai's path decomposition conjecture holds for the cartesian product $G\Box H$, where $H$ is any graph and $G$ is a unicyclic graph or a bicyclic graph.

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Xiaohong Chen, Baoyindureng Wu. 2023-10-18. Gallai's path decomposition conjecture for cartesian product of graphs (\uppercase\expandafter{\romannumeral 2}). https://arxiv.org/abs/2310.11704

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