Order 14 is the largest order for which every 4-total coloring of every cubic graph is equitable

Abstract

A total coloring of a graph is an assignment of colors to its vertices and edges so that adjacent or incident elements receive distinct colors, and it is equitable when the cardinalities of any two color classes differ by at most one. Stemock conjectured that every $4$-total coloring of a cubic graph of order less than $20$ is equitable. In this paper, we disprove this conjecture: the circular ladder $L_{12}$ admits a non-equitable $4$-total coloring and, moreover, no smaller counterexample exists: order $4$ is vacuous, and every $4$-total coloring of a cubic graph of order $6$, $8$, or $10$ is equitable. We also prove that the same property holds at order $14$. Our proofs rely on a decomposition lemma, which states that, in any $4$-total coloring of a cubic graph $G$, each color class consists of an independent set $S$ together with a perfect matching of $G-S$. We use the lemma to determine all possible color class configurations for orders $12$, $16$, and $18$, and we show that every listed configuration is attained. Finally, we provide a splicing construction showing that, for every even $n\geq16$, some connected cubic graph of order $n$ admits a non-equitable $4$-total coloring. We may conclude that $14$ is the largest order for which every $4$-total coloring of every cubic graph is equitable.