On multipartite derangement graphs

Andriaherimanana Sarobidy Razafimahatratra

Abstract


Given a finite transitive permutation group $G\leq \sym(\Omega)$, with $|\Omega|\geq 2$, the derangement graph $\Gamma_G$ of $G$ is the Cayley graph $\operatorname{Cay}(G,\Der(G))$, where $\Der(G)$ is the set of all derangements of $G$. Meagher et al. [On triangles in derangement graphs, {\it J. Combin. Theory, Ser. A}, 180:105390, 2021] recently proved that $\sym(2)$ acting on $\{1,2\}$ is the only transitive group whose derangement graph is bipartite and any transitive group of degree at least three has a triangle in its derangement graph. They also showed that there exist transitive groups whose derangement graphs are complete multipartite.

This paper gives two new families of transitive groups with complete multipartite derangement graphs. In addition, we construct an infinite family of transitive groups whose derangement graphs are multipartite but not complete multipartite.


Keywords


Derangement graph, cocliques, Erdős-Ko-Rado theorem, Cayley graphs

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DOI: https://doi.org/10.26493/1855-3974.2554.856

ISSN: 1855-3974

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