Cubic graphs attaining non-equitable feasible profiles

Characterize, for each even order n at least 20 and each feasible vertex profile with v_1-v_4 at least 4, the cubic graphs of order n that attain that profile.

Background

The paper defines a feasible profile of order n as a non-increasing quadruple (v_1,v_2,v_3,v_4) of nonnegative even integers summing to n and satisfying the counting constraints imposed by the decomposition of a 4-total coloring into independent vertex classes and perfect matchings. A profile with v_1-v_4 at least 4 corresponds to a non-equitable 4-total coloring.

Complete profile analyses are provided for orders 12, 16, and 18, and the paper constructs non-equitable colorings for every even order at least 16. The stated problem asks for a graph-theoretic characterization, for every even order at least 20, of precisely which cubic graphs realize each such non-equitable feasible profile.

References

Finally, in support of continued pursuit of the subject, we offer the following problem, which extends the complete analyses of orders $12$, $16$, and $18$ in Theorems~\ref{thm:n12}--\ref{thm:n18}. Characterize, for each even $n\geq20$ and each feasible profile with $v_1-v_4\geq4$, the cubic graphs of order $n$ that attain it.

— Order 14 is the largest order for which every 4-total coloring of every cubic graph is equitable  (2609.05259 - Adauto et al., 4 Sep 2026) in Section 6, “Final remarks,” immediately following Question 1