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arXiv · 2308.07684

Transitive path decompositions of Cartesian products of complete graphs

Ajani De Vas Gunasekara·Alice Devillers
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Abstract

An $H$-decomposition of a graph $\Gamma$ is a partition of its edge set into subgraphs isomorphic to $H$. A transitive decomposition is a special kind of $H$-decomposition that is highly symmetrical in the sense that the subgraphs (copies of $H$) are preserved and transitively permuted by a group of automorphisms of $\Gamma$. This paper concerns transitive $H$-decompositions of the graph $K_n \Box K_n$ where $H$ is a path. When $n$ is an odd prime, we present a construction for a transitive path decomposition where the paths in the decomposition are considerably large compared to the number of vertices. Our main result supports well-known Gallai's conjecture and an extended version of Ringel's conjecture.

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Ajani De Vas Gunasekara, Alice Devillers. 2023-08-15. Transitive path decompositions of Cartesian products of complete graphs. https://doi.org/10.1007/s10623-024-01493-9

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