13 research outputs found
On the Representability of Line Graphs
A graph G=(V,E) is representable if there exists a word W over the alphabet V
such that letters x and y alternate in W if and only if (x,y) is in E for each
x not equal to y. The motivation to study representable graphs came from
algebra, but this subject is interesting from graph theoretical, computer
science, and combinatorics on words points of view. In this paper, we prove
that for n greater than 3, the line graph of an n-wheel is non-representable.
This not only provides a new construction of non-representable graphs, but also
answers an open question on representability of the line graph of the 5-wheel,
the minimal non-representable graph. Moreover, we show that for n greater than
4, the line graph of the complete graph is also non-representable. We then use
these facts to prove that given a graph G which is not a cycle, a path or a
claw graph, the graph obtained by taking the line graph of G k-times is
guaranteed to be non-representable for k greater than 3.Comment: 10 pages, 5 figure
Restricted non-separable planar maps and some pattern avoiding permutations
Tutte founded the theory of enumeration of planar maps in a series of papers
in the 1960s. Rooted non-separable planar maps are in bijection with
West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori,
Jacquard and Schaeffer in 1997, as well as a family of permutations defined by
the avoidance of two four letter patterns. In this paper we give upper and
lower bounds on the number of multiple-edge-free rooted non-separable planar
maps. We also use the bijection between rooted non-separable planar maps and a
certain class of permutations, found by Claesson, Kitaev and Steingrimsson in
2009, to show that the number of 2-faces (excluding the root-face) in a map
equals the number of occurrences of a certain mesh pattern in the permutations.
We further show that this number is also the number of nodes in the
corresponding beta(1,0)-tree that are single children with maximum label.
Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure
Some properties of abelian return words (long abstract)
We investigate some properties of abelian return words as recently introduced by Puzynina and Zamboni. In particular, we obtain a characterization of Sturmian words with non-null intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the 2-automatic Thue–Morse word. We also investigate the relationship existing between abelian complexity and finiteness of the set of abelian returns to all prefixes. We end this paper by considering the notion of abelian derived sequence. It turns out that, for the Thue–Morse word, the set of abelian derived sequences is infinite
Avoiding 2-binomial squares and cubes
Two finite words are -binomially equivalent if, for all words of length at most , the number of occurrences of as a (scattered) subword of is equal to the number of occurrences of in . This notion is a refinement of the usual abelian equivalence. A -binomial square is a word where and are -binomially equivalent.
In this paper, considering pure morphic words, we prove that -binomial squares (resp. cubes) are avoidable over a -letter (resp. -letter) alphabet. The sizes of the alphabets are optimal
On uniform recurrence of a direct product
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to ApplicationsThe direct product of two words is a naturally defined word on the alphabet of pairs of symbols. An infinite word is uniformly recurrent if each its subword occurs in it with bounded gaps. An infinite word is strongly recurrent if the direct product of it with each uniformly recurrent word is also uniformly recurrent. We prove that fixed points of the expanding binary symmetric morphisms are strongly recurrent. In particular, such is the Thue-Morse word
On the number of abelian bordered words (with an example of automatic theorem-proving)
In the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a -letter alphabet, the connection with paths in the triangular lattice is made. Finally, we characterize the lengths of the abelian unbordered factors occurring in the Thue--Morse word using some kind of automatic theorem-proving provided by a logical characterization of the -automatic sequences
Another Generalization of Abelian Equivalence: Binomial Complexity of Infinite Words (long version)
The binomial coefficient of two words and is the number of times occurs as a subsequence of . Based on this classical notion, we introduce the -binomial equivalence of two words refining the abelian equivalence. Two words and are -binomially equivalent, if, for all words of length at most , the binomial coefficients of and and respectively, and are equal. The -binomial complexity of an infinite word maps an integer to the number of -binomial equivalence classes of factors of length
occurring in . We study the first properties of -binomial equivalence. We compute the -binomial complexity of two classes of words: Sturmian words and (pure) morphic words that are fixed points of Parikh-constant morphisms like the Thue--Morse word, i.e., images by the morphism of all the letters have the same Parikh vector. We prove that the frequency of each symbol of an infinite recurrent word with bounded -binomial complexity is rational
On the Number of Abelian Bordered Words
peer reviewedIn the literature, many bijections between (labeled) Motzkin paths and various other combinatorial objects are studied. We consider abelian (un)bordered words and show the connection with irreducible symmetric Motzkin paths and paths in Z not returning to the origin. This study can be extended to abelian unbordered words over an arbitrary alphabet and we derive expressions to compute the number of these words. In particular, over a 3-letter alphabet, the connection with paths in the triangular lattice is made. Finally, we study the lengths of the abelian unbordered factors occurring in the Thue--Morse word
Word-representability of line graphs
A graph G=(V,E) is representable if there exists a word W over the alphabet V such that letters x and y alternate in W if and only if (x ,y) is in E for each x not equal to y . The motivation to study representable graphs came from algebra, but this subject is interesting from graph theoretical, computer science, and combinatorics on words points of view. In this paper, we prove that for n greater than 3, the line graph of an n-wheel is non-representable. This not only provides a new construction of non-repre- sentable graphs, but also answers an open question on representability of the line graph of the 5-wheel, the minimal non-representable graph. Moreover, we show that for n greater than 4, the line graph of the complete graph is also non-representable. We then use these facts to prove that given a graph G which is not a cycle, a path or a claw graph, the graph obtained by taking the line graph of G k-times is guaranteed to be non-representable for k greater than 3