Unavoidable pattern

In mathematics and theoretical computer science, a pattern(term) is a sequence of characters over some alphabet.

A word $x\in \Sigma ^{*}$ is an instance of pattern $p\in \Delta ^{*}$, if there exists a non-erasing semigroup morphism $f:\Delta ^{*}\rightarrow \Sigma ^{*}$ such that $f(p)=x$. Non-erasing means $f(a)\neq \varepsilon ,\forall a\in \Delta$.

A word $w$ matches(encounters) pattern $p$ if there exists a factor of $w$, which is also an instance of $p$. Otherwise, $w$ avoids $p$. If $w$ avoids $p$, $w$ is also called $p$-free.

A pattern $p$ is unavoidable on a finite alphabet $\Sigma$ if there exists $n\in \mathrm {N}$, word $x$ matches $p$ for all $x\in \Sigma ^{*},|x|\geq n$. Otherwise, $p$ is avoidable on $\Sigma$, which implies there exist infinitely many words over alphabet $\Sigma$ that avoid $p$. This is equivalent to saying that $p$ is avoidable on $\Sigma$ if there exists an infinite word $w\in \Sigma ^{\omega }$ that avoids $p$.

A pattern $p$ is an unavoidable pattern(blocking term) if $p$ is unavoidable on any finite alphabet.

When a pattern $p$ is said to be unavoidable and does not specify an alphabet, then $p$ is unavoidable for any finite alphabet. Conversely, a pattern $p$ is said to be avoidable if $p$ is avoidable on some finite alphabet.

Properties

• Let pattern $q$ be an instance of avoidable pattern $p$, then $q$ is also avoidable.
• Let avoidable pattern $p$ be a factor of pattern $q$, then $q$ is also avoidable.
• A pattern $q$ is unavoidable if and only if $q$ is a factor of some unavoidable pattern $p$.
• Given an unavoidable pattern $p$ and a character $a$ not in $p$, then $pap$ is unavoidable.
• Given an unavoidable pattern $p$, there exist a character $a$ such that $a$ occurs excatly once in $p$.
• Given a pattern $p$, let $n$ represents the number of distinct characters of $p$. if $|p|\geq 2^{n}$, then $p$ is avoidable.

Zimin words

Main article: Sesquipower

Given alphabet $\Delta =\{x_{1},x_{2},...\}$, Zimin words(patterns) are defined recursively $Z_{n+1}=Z_{n}x_{n+1}Z_{n}$ for $n\in \mathrm {Z} ^{+}$ and base case $Z_{1}=x_{1}$.

All Zimin words are unavoidable.

A word $w$ is unavoidable if and only if it's a factor of a Zimin word.

Given a finite alphabet $\Sigma$, let $f(n,|\Sigma |)$ represents the smallest $m\in \mathrm {Z} ^{+}$ such that $w$ matches $Z_{n}$ for all $w\in \Sigma ^{m}$. We have following properties:

• $f(1,q)=1$
• $f(2,q)=2q+1$
• $f(3,2)=29$
• $f(n,q)\leq {^{n-1}q}=\underbrace {q^{q^{\cdot ^{\cdot ^{q}}}}} _{n-1}$

$Z_{n}$ is the longest unavoidable patterns constructed by alphabet $\Delta =\{x_{1},x_{2},...,x_{n}\}$ since $|Z_{n}|=2^{n}-1$

Pattern reduction

Given a pattern $p$ over some alphabet $\Delta$, we say $x\in \Delta$ is free for $p$ if there exist subsets $A,B$ of $\Delta$ such that the following hold:

1. $uv$ is a factor of $p$ and $u\in A$ ↔  $uv$ is a factor of $p$ and $v\in B$
2. $x\in A\backslash B\cup B\backslash A$

e.g. let $p=abcbab$, then $b$ is free for $p$ since there exist $A=ac,B=b$ satisfied conditions above.

A pattern $p\in \Delta ^{*}$ can reduce to pattern $q$ if there exists a character $x\in \Delta$ such that $x$ is free for $p$, and $q$ can be obtained by removing all occurrence of $x$ from $p$. Denote as $p{\stackrel {x}{\rightarrow }}q$.

e.g. let $p=abcbab$, then $p$ can reduce to $q=aca$ since $b$ is free for $p$.

Given patterns $p,q,r$, if $p$ can reduce to $q$ and $q$ can reduce to $r$, then $p$ can reduce to $r$. Denote as $p{\stackrel {*}{\rightarrow }}r$.

A pattern $p$ is unavoidable if and only if $p$ can reduce to the empty word, hence $p{\stackrel {*}{\rightarrow }}\varepsilon$.

Avoidability index

A pattern $p$ is $k$-avoidable if $p$ is avoidable on an alphabet $\Sigma$ of size $k$. Otherwise, $p$ is $k$-unavoidable, which means $p$ is unavoidable on every alphabet of size $k$.

if pattern $p$ is $k$-avoidable, then $p$ is $g$-avoidable for all $g\geq k$.

The avoidability index of a pattern $p$ is the smallest $k$ such that $p$ is $k$-avoidable, $\infty$ if $p$ is unavoidable.

Graph pattern avoidance

Given a simple graph $G=(V,E)$, a coloring $c:E\rightarrow \Delta$ matches pattern $p$ if there exists a simple path $P=[e_{1},e_{2},...,e_{r}]$ in $G$ such that the sequence $c(P)=[c(e_{1}),c(e_{2}),...,c(e_{r})]$ matches $p$. Otherwise, $c$ is said to avoid $p$ or $p$-free.

The pattern chromatic number $\pi _{p}(G)$ is the minimal number of distinct colors needed for a $p$-free coloring $c$ over the graph $G$.

Let $\pi _{p}(n)=max\{\pi _{p}(G):G\in G_{n}\}$ where $G_{n}$ is the set of all simple graphs with a maximum degree no more than $n$.

A pattern $p$ is avoidable on graphs if $\pi _{p}(n)$ is bounded by $c_{p}$, where $c_{p}$ only depend on $p$.

• Avoidance on words can be expressed as a specific case of avoidance on graphs, hence a pattern $p$ is avoidable on any finite alphabet if and only if $\pi _{p}(P_{n})\leq c_{p}$ for all $n\in \mathrm {Z} ^{+}$ where $P_{n}$ is a graph of $n$ vertexs concatenated.

The minimum multiplicity of pattern $p$ is $m(p)=min(count_{p}(x):x\in p)$ where $count_{p}(x)$ is the number of occurance of character $x$ in pattern $p$.

There exist an absolute constant $c$, such that $\pi _{p}(n)\leq cn^{\frac {m(p)}{m(p)-1}}$for all pattern $p$ with $m(p)\geq 2$

Given a pattern $p$, let $n$ represents the number of distinct characters of $p$. if $|p|\geq 2^{n}$, then $p$ is avoidable on graphs.

Examples

• The Thue–Morse sequence is cube-free and overlap-free, hence it avoids the patterns $aaa$ and $ababa$.
• The patterns $a$ and $aba$ are unavoidable on any alphabet, since they are factors of the Zimin words.
• The power patterns $x^{n}$ for $n\geq 3$ are 2-avoidable.
• A square-free word is one avoiding the pattern $xx$. An example is the word over the alphabet $\{0,\pm 1\}$ obtained by taking the first difference of the Thue–Morse sequence. See also Square-free word.
• all binary patterns can divided into three categories:
  • $\varepsilon ,a,ab,aba$ are unavoidable.
  • $aa,aab,abb,aaba,aabb,abaa,abab,abba,aabaa,aabab,ababb$ have avoidability index of 3.
  • others have avoidability index of 2.
• abwbaxbcyaczca has avoidability index of 4, as well as other locked words. (Baker, McNulty, Taylor 1989)
• abvbawbcxacycdazdcd has avoidability index of 5. (Clark 2004)

Open problems

• Is there an avoidable pattern $p$ such that the avoidability index of $p$ is 6?

References

• Allouche, Jean-Paul; Shallit, Jeffrey (2003). Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press. ISBN 978-0-521-82332-6. Zbl 1086.11015.
• Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. Vol. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
• Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
• Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.