# Chordality of enhanced power graphs of simple groups

- Reference: `kourovka-21-23-chordality-of-enhanced-power-graphs-of-simple-groups`
- Page: https://theoremdb.org/statements/kourovka-21-23-chordality-of-enhanced-power-graphs-of-simple-groups
- Record maturity: Reviewed problem

## Problem

A graph is called a cograph if it has no induced subgraph isomorphic to a path with 4 vertices. A graph is said to be chordal if it has no induced cycles with n vertices for every n ⩾ 4. For a finite group G, the enhanced power graph E(G) is the graph with vertex set G and edges {x, y} for all x ̸= y ∈ G such that ⟨x, y⟩ is cyclic. (a) For a given integer n ⩾ 4, determine the set of all finite nonabelian simple groups G such that E(G) has no induced cycles with n vertices. (b) Determine the set of all finite nonabelian simple groups G such that E(G) is chordal. In (Preprint, 2025, https://arxiv.org/abs/2510.18073) we proved that if the enhanced power graph of a given finite group is a cograph, then it is also chordal. Also the finite nonabelian simple groups whose enhanced power graph is a cograph are described, and additional information is obtained on finite nonabelian simple groups whose enhanced power graph has no induced cycles with 4 vertices. This is Kourovka Notebook Problem \(21.23\).

## Status

The reviewed record remains open.

## Work

### Working on this

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## References

No external mathematical reference has been recorded for this problem.