# On some interesting ternary formulas

## TL;DR

This paper investigates the avoidance of specific ternary formulas in words and graphs, revealing new avoidability results, counterexamples to conjectures, and classifications of formulas based on their avoidability index.

## Contribution

It provides new classifications of ternary formulas' avoidability, disproves a conjecture, and links pattern avoidance to graph properties.

## Key findings

- Only certain infinite ternary words avoid specific formulas.
- Some formulas are avoided by polynomially many binary words.
- The pattern ABACADABCA is unavoidable in certain graph classes.

## Abstract

We obtain the following results about the avoidance of ternary formulas. Up to renaming of the letters, the only infinite ternary words avoiding the formula $ABCAB.ABCBA.ACB.BAC$ (resp. $ABCA.BCAB.BCB.CBA$) have the same set of recurrent factors as the fixed point of $\texttt{0}\mapsto\texttt{012}$, $\texttt{1}\mapsto\texttt{02}$, $\texttt{2}\mapsto\texttt{1}$. The formula $ABAC.BACA.ABCA$ is avoided by polynomially many binary words and there exist arbitrarily many infinite binary words with different sets of recurrent factors that avoid it. If every variable of a ternary formula appears at least twice in the same fragment, then the formula is $3$-avoidable. The pattern $ABACADABCA$ is unavoidable for the class of $C_4$-minor-free graphs with maximum degree~$3$. This disproves a conjecture of Grytczuk. The formula $ABCA.ACBA$, or equivalently the palindromic pattern $ABCADACBA$, has avoidability index $4$.