Binary words containing infinitely many overlaps

On the context-freeness of the set of words containing overlaps

We show that the set of binary words containing overlaps is not unambiguously context-free and that the set of ternary words containing overlaps is not context-free. We also show that the set of binary words that are not subwords of the Thue-Morse word is not unambiguously context-free.

Morphisms generating antipalindromic words

We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism $\mathrm{E}:\{0,1\}^*\to\{0,1\}^*$, defined by $\mathrm{E}(w_1\cdots w_n)=(1-w_{n})\cdots(1-w_1)$. We conjecture that these two classes contain all morphisms (up to conjugation) which generate infinite words with infinitely many antipalindromes. This is an analogue to the famous HKS conjecture concerning infinite words containing infinitely many palindromes. We prove our conjectu…

十个正方形强制重叠

Ten Squares Force an Overlap

本文证明任意10个或更多二进制正方形的拼接必然包含重叠，且10是最优上界，而三元字母表上存在无限长的无重叠词由正方形拼接而成。

A Note on Squares in Binary Words

We consider words over a binary alphabet. A word $w$ is overlap-free if it does not have factors (blocks of consecutive letters) of the form $uvuvu$ for nonempty $u$. Let $M(w)$ denote the number of positions that are middle positions of squares in $w$. We show that for overlap-free binary words, $2M(w) \le |w|+3$, and that there are infinitely many overlap-free binary words for which $2M(w)=|w|+3$.

Avoidance of split overlaps

We generalize Axel Thue's familiar definition of overlaps in words, and show that there are no infinite words containing split occurrences of these generalized overlaps. Along the way we prove a useful theorem about repeated disjoint occurrences in words -- an interesting natural variation on the classical de Bruijn sequences.

有向图中的无限网格

Infinite grids in digraphs

本文研究了有向图中无限网格的结构，证明了具有无限多个互不相交有向射线的端点的有向图包含特定的网格状子图。