Word-representability of 6- and 7-regular circulant graphs

Determine whether all 6-regular and all 7-regular circulant graphs are word-representable, or construct counterexamples in either class.

Background

The paper recalls a known construction of a non-word-representable 8-regular circulant graph, namely C_{14}(3,4,5,6), obtained by setting n=14 and r=3 in an earlier theorem. This motivates asking whether lower regularities exhibit universal word-representability or whether analogous counterexamples already occur for 6-regular or 7-regular circulant graphs. The problem explicitly leaves both alternatives open.

References

A natural problem is to determine whether all $6$-regular and $7$-regular circulant graphs are word-representable, or whether these classes also contain counterexamples.

— On the Word-Representability of 5-Regular Circulant Graphs  (2512.05480 - Roy et al., 5 Dec 2025) in Problem, Section 4 (Conclusion and Future Work), Section \ref{con}