We study infinite ternary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponent.

A word contains a half-flip if it contains non-empty factors uv and vu where |u|=|v|. Fici reports a non-constructive proof of the existence of an infinite word over a finite alphabet avoiding half-flips and asks for the size of the smallest alphabet over which half-flips may be avoided. Currie and Rampersad have proposed a pure morphic word over 8 letters and a morphic word over 5 letters and conjecture that they avoid half-flips. We present a pure morphic word over 5 letters that avoids half-flips. We also show that half-flips with |u|≥2 are 3-avoidable and that half-flips with |u|≥4 are 2-avoidable.

An interesting phenomenon in combinatorics on words is when every recurrent word satisfying some avoidance constraints has the same factor set as a morphic word. An early example is the Hall-Thue word, fixed point of the morphism 0-012, 1-02, 2-1, which is essentially the only ternary word avoiding squares and the factors 010 and 212. We provide some examples of this phenomenon from various contexts.

We study a question of Harju from 2019 regarding the existence of infinite ternary square-free words whose subsequences modulo p and q are also square-free for relatively prime integers p and q. Among such pairs (p, q) with p, q ≥ 3, the only two pairs with this property known prior to this work were (3, 11) and (5, 6). We prove that there are finitely many pairs (p, q) of relatively prime integers with p, q ≥ 3 for which there is no infinite ternary square-free word whose subsequences modulo p and q are square-free. To prove our result, we combine different techniques, including the construction of words from multi-valued square-free morphisms and circular square-free morphisms. We also introduce the notion of square-free transducers, a generalization of square-free morphisms that may be of independent interest.

We say that a finite factor f of a word w is imaged if there exists a non-erasing morphism m m, distinct from the identity, such that w contains m(f). We show that every infinite word contains an imaged factor of length at least 6 and that 6 is best possible. We show that every infinite binary word contains at least 36 distinct imaged factors and that 36 is best possible.

The complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. We study infinite binary words $\bf w$ that avoid sufficiently large complementary factors; that is, if $x$ is a factor of $\bf w$ then $\overline{x}$ is not a factor of $\bf w$. In particular, we classify such words according to their critical exponents.

A tangram is a word in which every letter occurs an even number of times. Thus it can be cut into parts that can be arranged into two identical words. The \emph{cut number} of a tangram is the minimum number of required cuts in this process. Tangrams with cut number one corresponds to squares. For $k\ge1$, let $t(k)$ denote the minimum size of an alphabet over which an infinite word avoids tangrams with cut number at most~$k$. The existence of infinite ternary square-free words shows that $t(1)=t(2)=3$. We show that $t(3)=t(4)=4$, answering a question from Dębski, Grytczuk, Pawlik, Przybyło, and Śleszyńska-Nowak.

A (proper) colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1,1)-Labelling). A classical complexity result on Colouring is a well-known dichotomy for H-free graphs (a graph is H-free if it does not contain H as an induced subgraph). In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We perform such a study and give almost complete complexity classifications for Acyclic Colouring, Star Colouring and Injective Colouring on H-free graphs (for each of the problems, we have one open case). Moreover, we give full complexity classifications if the number of colours k is fixed, that is, not part of the input. From our study it follows that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs of multigraphs.

We study infinite binary words that contain few distinct palindromes. In particular, we classify such words according to their critical exponents. This extends results by Fici and Zamboni [TCS 2013]. Interestingly, the words with 18 and 20 palindromes happen to be morphic images of the fixed point of the morphism $\texttt{0}\mapsto\texttt{01}$, $\texttt{1}\mapsto\texttt{21}$, $\texttt{2}\mapsto\texttt{0}$.

We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete.

In combinatorics on words, a word $w$ over an alphabet $\Sigma$ is said to avoid a pattern $p$ over an alphabet $\Delta$ if there is no factor $f$ of $w$ such that $f=h(p)$ where $h:\Delta^*\to\Sigma^*$ is a non-erasing morphism. A pattern $p$ is said to be $k$-avoidable if there exists an infinite word over a $k$-letter alphabet that avoids $p$. A pattern is \emph{doubled} if every variable occurs at least twice. Doubled patterns are known to be $3$-avoidable. Currie, Mol, and Rampersad have considered a generalized notion which allows variable occurrences to be reversed. That is, $h(V^R)$ is the mirror image of $h(V)$ for every $V\in\Delta$. We show that doubled patterns with reversal are $3$-avoidable. We also conjecture that (classical) doubled patterns that do not contain a square are $2$-avoidable. We confirm this conjecture for patterns with at most 4 variables. This implies that for every doubled pattern $p$, the growth rate of ternary words avoiding $p$ is at least the growth rate of ternary square-free words. A previous version of this paper containing only the first result has been presented at WORDS 2021.

We study the properties of the ternary infinite word p = 012102101021012101021012⋯, that is, the fixed point of the map h : 0 → 01, 1 → 21, 2 → 0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We compute its abelian complexity and determine the lengths of its bispecial factors. Finally, we give a characterization of p in terms of avoided factors.

Thue has shown the existence of three types of infinite square-free words over {0,1,2} avoiding the factor 010. They respectively avoid {010,212}, {010,101}, and {010,020}. Also Dejean constructed an infinite $\left(\tfrac74^+\right)$-free ternary word. A word is $d$-directed if it does not contain both a factor of length $d$ and its mirror image. We show that there exist exponentially many $\left(\tfrac74^+\right)$-free 180-directed ternary words avoiding 010. Moreover, there does not exist an infinite $\left(\tfrac74^+\right)$-free 179-directed ternary word avoiding 010.

This paper is devoted to the study of particular geometrically defined intersection classes of graphs. Those were previously studied by Magnant and Martin, who proved that these graphs have arbitrary large chromatic number, while being triangle-free. We give several structural properties of these graphs, and we raise several questions.

The (bitwise) complement $\overline{x}$ of a binary word $x$ is obtained by changing each $0$ in $x$ to $1$ and vice versa. An $\textit{antisquare}$ is a nonempty word of the form $x\, \overline{x}$. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is $(5+\sqrt{5})/2$. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is $\textit{good}$ if the only antisquares it contains are $01$ and $10$. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length $n$ and determine the repetition threshold between polynomial and exponential growth for the number of good words.

We consider sets of factors that can be avoided in square-free words on two-generator free groups. The elements of the group are presented in terms of {0, 1, 2, 3} such that 0 and 2 (resp.,1 and 3) are inverses of each other so that 02, 20, 13 and 31 do not occur in a reduced word. A Dean word is a reduced word that does not contain occurrences of uu for any nonempty u. Dean showed in 1965 that there exist infinite square-free reduced words. We show that if w is a Dean word of length at least 59 then there are at most six reduced words of length 3 avoided by w. We construct an infinite Dean word avoiding six reduced words of length 3. We also construct infinite Dean words with low critical exponent and avoiding fewer reduced words of length 3. Finally, we show that the minimal frequency of a letter in a Dean word is 8 59 and the growth rate is close to 1.45818.

We characterize the formulas that are avoided by every αfree word for some α > 1. We study the avoidability index of formulas whose fragments are of the form XYX. The largest avoidability index of an avoidable palindrome pattern is known to be at least 4 and at most 16. We make progress toward the conjecture that every avoidable palindrome pattern is 4-avoidable.

In combinatorics on words, a word w over an alphabet \(\varSigma \) is said to avoid a pattern p over an alphabet \(\varDelta \) if there is no factor f of w such that \(f=h(p)\) where \(h:\varDelta ^*\rightarrow \varSigma ^*\) is a non-erasing morphism. A pattern p is said to be k-avoidable if there exists an infinite word over a k-letter alphabet that avoids p. A pattern is doubled if every variable occurs at least twice. Doubled patterns are known to be 3-avoidable. Currie, Mol, and Rampersad have considered a generalized notion which allows variable occurrences to be reversed. That is, \(h(V^R)\) is the mirror image of h(V) for every \(V\in \varDelta \). We show that doubled patterns with reversal are 3-avoidable.

We present a family of avoidable formulas with reversal whose avoidability index is unbounded. We also complete the determination of the avoidability index of the formulas with reversal in the 3-avoidance basis.

We characterize the formulas that are avoided by every $\alpha$-free word for some $\alpha>1$. We show that the avoidable formulas whose fragments are of the form $XY$ or $XYX$ are $4$-avoidable. The largest avoidability index of an avoidable palindrome pattern is known to be at least $4$ and at most $16$. We make progress toward the conjecture that every avoidable palindrome pattern is $4$-avoidable.