Equitable 4-total colorings of small Type 1 cubic graphs

Determine whether every Type 1 cubic graph of order less than 20 admits at least one equitable 4-total coloring, equivalently whether the graph R is a Type 1 cubic graph with equitable total chromatic number 5 of minimum order.

Background

The paper studies equitable 4-total colorings of cubic graphs, in which the four color classes have cardinalities differing by at most one. It establishes that non-equitable 4-total colorings occur at orders 12, 16, and 18, while also showing that the specific counterexamples at those orders—namely the circular ladders L_12 and L_18 and the graphs H_16 and H_18—admit equitable 4-total colorings as well.

The graph R, previously known from the literature, is Type 1 but admits no equitable 4-total coloring, so its equitable total chromatic number is 5 despite its total chromatic number being 4. The question asks whether a Type 1 cubic graph with this stronger obstruction exists at any smaller order below 20. The authors note that the question can be checked over the finite catalogue of connected cubic graphs of orders at most 18, potentially using integer programming or SAT.