arXiv subjects

Sergey Kitaev

Publications and source records attributed to Sergey Kitaev.

At least 91 records · Page 5Linked to original sources

arXiv2015

Word-representability of triangulations of grid-covered cylinder graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$, $x\neq y$, alternate in $w$ if and only if $(x,y)\in E$. Halldórsson et al.\ have shown that a graph is word-representable if and only if it admits a so-called semi-transitive orientation. A corollary to this result is that any 3-colorable graph is word-representable. Akrobotu et al.\ have shown that a triangulation of a grid graph is word-representable if and only if it is 3-colorable. This result does not hold for triangulations of grid-covered cylinder graphs, namely, there are such word-representable graphs with chromatic number 4. In this paper we show that word-representability of triangulations of grid-covered cylinder graphs with three sectors (resp., more than three sectors) is characterized by avoiding a certain set of six minimal induced subgraphs (resp., wheel graphs $W_5$ and $W_7$).

math.CO↗

arXiv2015

Existence of $u$-representation of graphs

Recently, Jones et al. introduced the study of $u$-representable graphs, where $u$ is a word over $\{1,2\}$ containing at least one 1. The notion of a $u$-representable graph is a far-reaching generalization of the notion of a word-representable graph studied in the literature in a series of papers. Jones et al. have shown that any graph is $11\cdots 1$-representable assuming that the number of 1s is at least three, while the class of 12-rerpesentable graphs is properly contained in the class of comparability graphs, which, in turn, is properly contained in the class of word-representable graphs corresponding to 11-representable graphs. Further studies in this direction were conducted by Nabawanda, who has shown, in particular, that the class of 112-representable graphs is not included in the class of word-representable graphs. Jones et al. raised a question on classification of $u$-representable graphs at least for small values of $u$. In this paper we show that if $u$ is of length at least 3 then any graph is $u$-representable. This rather unexpected result shows that from existence of representation point of view there are only two interesting non-equivalent cases in the theory of $u$-representable graphs, namely, those of $u=11$ and $u=12$.

math.CO↗

arXiv2015

Pattern-avoiding alternating words

A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1 w_3 \cdots$ (when the word is up-down) or $w_1>w_2 w_4<\cdots$ (when the word is down-up). In this paper, we initiate the study of (pattern-avoiding) alternating words. We enumerate up-down (equivalently, down-up) words via finding a bijection with order ideals of a certain poset. Further, we show that the number of 123-avoiding up-down words of even length is given by the Narayana numbers, which is also the case, shown by us bijectively, with 132-avoiding up-down words of even length. We also give formulas for enumerating all other cases of avoidance of a permutation pattern of length 3 on alternating words.

math.CO↗

arXiv2015

Word-representability of subdivisions of triangular grid graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$. A triangular grid graph is a subgraph of a tiling of the plane with equilateral triangles defined by a finite number of triangles, called cells. A subdivision of a triangular grid graph is replacing some of its cells by plane copies of the complete graph $K_4$. Inspired by a recent elegant result of Akrobotu et al., who classified word-representable triangulations of grid graphs related to convex polyominoes, we characterize word-representable subdivisions of triangular grid graphs. A key role in the characterization is played by smart orientations introduced by us in this paper. As a corollary to our main result, we obtain that any subdivision of boundary triangles in the Sierpiński gasket graph is word-representable.

math.CO↗

arXiv2015

p-Ascent Sequences

A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is an {\em ascent sequence} if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most 1 plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes, and Kitaev, who showed that these sequences of length $n$ are in 1-to-1 correspondence with \tpt-free posets of size $n$, which, in turn, are in 1-to-1 correspondence with interval orders of size $n$. Ascent sequences are also in bijection with several other classes of combinatorial objects including the set of upper triangular matrices with nonnegative integer entries such that no row or column contains all zeros, permutations that avoid a certain mesh pattern, and the set of Stoimenow matchings. In this paper, we introduce a generalization of ascent sequences, which we call {\em $p$-ascent sequences}, where $p \geq 1$. A sequence $(a_1, \ldots, a_n)$ of nonnegative integers is a $p$-ascent sequence if $a_0 =0$ and for all $i \geq 2$, $a_i$ is at most $p$ plus the number of ascents in $(a_1, \ldots, a_{i-1})$. Thus, in our terminology, ascent sequences are 1-ascent sequences. We generalize a result of the authors by enumerating $p$-ascent sequences with respect to the number of $0$s. We also generalize a result of Dukes, Kitaev, Remmel, and Steingr\'ımsson by finding the generating function for the number of $p$-ascent sequences which have no consecutive repeated elements. Finally, we initiate the study of pattern-avoiding $p$-ascent sequences.

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arXiv2015

Semi-Transitive Orientations and Word-Representable Graphs

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.

math.CO↗

arXiv2014

Representing Graphs via Pattern Avoiding Words

The notion of a word-representable graph has been studied in a series of papers in the literature. A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $xy$ is an edge in $E$. If $V =\{1, \ldots, n\}$, this is equivalent to saying that $G$ is word-representable if for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if the subword $w_{\{x,y\}}$ of $w$ consisting of all occurrences of $x$ or $y$ in $w$ has no consecutive occurrence of the pattern 11. In this paper, we introduce the study of $u$-representable graphs for any word $u \in \{1,2\}^*$. A graph $G$ is $u$-representable if and only if there is a labeled version of $G$, $G=(\{1, \ldots, n\}, E)$, and a word $w \in \{1, \ldots, n\}^*$ such that for all $x,y \in \{1, \ldots, n\}$, $xy \in E$ if and only if $w_{\{x,y\}}$ has no consecutive occurrence of the pattern $u$. Thus, word-representable graphs are just $11$-representable graphs. We show that for any $k \geq 3$, every finite graph $G$ is $1^k$-representable. This contrasts with the fact that not all graphs are 11-representable graphs. The main focus of the paper is the study of $12$-representable graphs. In particular, we classify the $12$-representable trees. We show that any $12$-representable graph is a comparability graph and the class of $12$-representable graphs include the classes of co-interval graphs and permutation graphs. We also state a number of facts on $12$-representation of induced subgraphs of a grid graph.

math.CO↗

arXiv2014

Number of cycles in the graph of 312-avoiding permutations

The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. That is, for every permutation $π= π_{1} π_{2} ... π_{n+1}$ there is a directed edge from the standardization of $π_{1} π_{2} ... π_{n}$ to the standardization of $π_{2} π_{3} ... π_{n+1}$. We give a formula for the number of cycles of length $d$ in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations and point out some open problems on this graph, which so far has been little studied.

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arXiv2014

Mahonian STAT on words

In 2000, Babson and Steingrímsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity. These conjectures were proved by Foata and Zeilberger in 2001, and by Foata and Randrianarivony in 2006. In 2010, Burstein refined some of these results by giving a bijection between permutations with a fixed value for the major index and those with the same value for STAT, where STAT is one of the statistics defined and proved to be Mahonian in the 2000 Babson and Steingrímsson's paper. Several other statistics are preserved as well by Burstein's bijection. At the Formal Power Series and Algebraic Combinatorics Conference (FPSAC) in 2010, Burstein asked whether his bijection has other interesting properties. In this paper, we not only show that Burstein's bijection preserves the Eulerian statistic ides, but also use this fact, along with the bijection itself, to prove Mahonity of the statistic STAT on words we introduce in this paper. The words statistic STAT introduced by us here addresses a natural question on existence of a Mahonian words analogue of STAT on permutations. While proving Mahonity of our STAT on words, we prove a more general joint equidistribution result involving two six-tuples of statistics on (dense) words, where Burstein's bijection plays an important role.

cs.DM↗

arXiv2014

Quadrant marked mesh patterns in 132-avoiding permutations I

This paper is a continuation of the systematic study of the distributions of quadrant marked mesh patterns initiated in [6]. Given a permutation $\sg = \sg_1 ... \sg_n$ in the symmetric group $S_n$, we say that $\sg_i$ matches the quadrant marked mesh pattern $MMP(a,b,c,d)$ if there are at least $a$ elements to the right of $\sg_i$ in $\sg$ that are greater than $\sg_i$, at least $b$ elements to left of $\sg_i$ in $\sg$ that are greater than $\sg_i$, at least $c$ elements to left of $\sg_i$ in $\sg$ that are less than $\sg_i$, and at least $d$ elements to the right of $\sg_i$ in $\sg$ that are less than $\sg_i$. We study the distribution of $MMP(a,b,c,d)$ in 132-avoiding permutations. In particular, we study the distribution of $MMP(a,b,c,d)$, where only one of the parameters $a,b,c,d$ are non-zero. In a subsequent paper [7], we will study the the distribution of $MMP(a,b,c,d)$ in 132-avoiding permutations where at least two of the parameters $a,b,c,d$ are non-zero.

math.CO↗

arXiv2014

On word-representability of polyomino triangulations

A graph $G=(V,E)$ is word-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 an edge in $E$. Some graphs are word-representable, others are not. It is known that a graph is word-representable if and only if it accepts a so-called semi-transitive orientation. The main result of this paper is showing that a triangulation of any convex polyomino is word-representable if and only if it is 3-colorable. We demonstrate that this statement is not true for an arbitrary polyomino. We also show that the graph obtained by replacing each $4$-cycle in a polyomino by the complete graph $K_4$ is word-representable. We employ semi-transitive orientations to obtain our results.

math.CO↗

arXiv2014

On graphs with representation number 3

A graph $G=(V,E)$ is word-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 an edge in $E$. A graph is word-representable if and only if it is $k$-word-representable for some $k$, that is, if there exists a word containing $k$ copies of each letter that represents the graph. Also, being $k$-word-representable implies being $(k+1)$-word-representable. The minimum $k$ such that a word-representable graph is $k$-word-representable, is called graph's representation number. Graphs with representation number 1 are complete graphs, while graphs with representation number 2 are circle graphs. The only fact known before this paper on the class of graphs with representation number 3, denoted by $\mathcal{R}_3$, is that the Petersen graph and triangular prism belong to this class. In this paper, we show that any prism belongs to $\mathcal{R}_3$, and that two particular operations of extending graphs preserve the property of being in $\mathcal{R}_3$. Further, we show that $\mathcal{R}_3$ is not included in a class of $c$-colorable graphs for a constant $c$. To this end, we extend three known results related to operations on graphs. We also show that ladder graphs used in the study of prisms are $2$-word-representable, and thus each ladder graph is a circle graph. Finally, we discuss $k$-word-representing comparability graphs via consideration of crown graphs, where we state some problems for further research.

math.CO↗

arXiv2014

S-crucial and bicrucial permutations with respect to squares

A permutation is square-free if it does not contain two consecutive factors of length two or more that are order-isomorphic. A permutation is bicrucial with respect to squares if it is square-free but any extension of it to the right or to the left by any element gives a permutation that is not square-free. Bicrucial permutations with respect to squares were studied by Avgustinovich et al., who proved that there exist bicrucial permutations of lengths $8k+1, 8k+5, 8k+7$ for $k\ge 1$. It was left as open questions whether bicrucial permutations of even length, or such permutations of length $8k+3$ exist. In this paper, we provide an encoding of orderings which allows us, using the constraint solver Minion, to show that bicrucial permutations of even length exist, and the smallest such permutations are of length 32. To show that 32 is the minimum length in question, we establish a result on left-crucial (that is, not extendable to the left) square-free permutations which begin with three elements in monotone order. Also, we show that bicrucial permutations of length $8k+3$ exist for $k=2,3$ and they do not exist for $k=1$. Further, we generalise the notions of right-crucial, left-crucial, and bicrucial permutations studied in the literature in various contexts, by introducing the notion of $P$-crucial permutations that can be extended to the notion of $P$-crucial words. In S-crucial permutations, a particular case of $P$-crucial permutations, we deal with permutations that avoid prohibitions, but whose extensions in any position contain a prohibition. We show that S-crucial permutations exist with respect to squares, and minimal such permutations are of length 17. Finally, using our software, we generate much of relevant data showing, for example, that there are 162,190,472 bicrucial square-free permutations of length 19.

math.CO↗

arXiv2014

New results on word-representable graphs

A graph $G=(V,E)$ is word-representable if there exists a word $w$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $w$ if and only if $(x,y)\in E$ for each $x\neq y$. The set of word-representable graphs generalizes several important and well-studied graph families, such as circle graphs, comparability graphs, 3-colorable graphs, graphs of vertex degree at most 3, etc. By answering an open question from [M. Halldorsson, S. Kitaev and A. Pyatkin, Alternation graphs, Lect. Notes Comput. Sci. 6986 (2011) 191--202. Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011, Tepla Monastery, Czech Republic, June 21-24, 2011.], in the present paper we show that not all graphs of vertex degree at most 4 are word-representable. Combining this result with some previously known facts, we derive that the number of $n$-vertex word-representable graphs is $2^{\frac{n^2}{3}+o(n^2)}$.

math.CO↗

arXiv2013

Frame patterns in n-cycles

In this paper, we study the distribution of the number of occurrences of the simplest frame pattern, called the $μ$ pattern, in $n$-cycles. Given an $n$-cycle $C$, we say that a pair $\langle i,j \rangle$ matches the $μ$ pattern if $i < j$ and as we traverse around $C$ in a clockwise direction starting at $i$ and ending at $j$, we never encounter a $k$ with $i < k < j$. We say that $ \langle i,j \rangle$ is a nontrivial $μ$-match if $i+1 < j$. Also, an $n$-cycle $C$ is incontractible if there is no $i$ such that $i+1$ immediately follows $i$ in $C$. We show that the number of incontractible $n$-cycles in the symmetric group $S_n$ is $D_{n-1}$, where $D_n$ is the number of derangements in $S_n$. Further, we prove that the number of $n$-cycles in $S_n$ with exactly $k$ $μ$-matches can be expressed as a linear combination of binomial coefficients of the form $\binom{n-1}{i}$ where $i \leq 2k+1$. We also show that the generating function $NTI_{n,μ}(q)$ of $q$ raised to the number of nontrivial $μ$-matches in $C$ over all incontractible $n$-cycles in $S_n$ is a new $q$-analogue of $D_{n-1}$, which is different from the $q$-analogues of the derangement numbers that have been studied by Garsia and Remmel and by Wachs. We show that there is a rather surprising connection between the charge statistic on permutations due to Lascoux and Schüzenberger and our polynomials in that the coefficient of the smallest power of $q$ in $NTI_{2k+1,μ}(q)$ is the number of permutations in $S_{2k+1}$ whose charge path is a Dyck path. Finally, we show that $NTI_{n,μ}(q)|_{q^{\binom{n-1}{2} -k}}$ and $NT_{n,μ}(q)|_{q^{\binom{n-1}{2} -k}}$ are the number of partitions of $k$ for sufficiently large $n$.

math.CO↗

arXiv2013

Universal graphs and universal permutations

Let $X$ be a family of graphs and $X_n$ the set of $n$-vertex graphs in $X$. A graph $U^{(n)}$ containing all graphs from $X_n$ as induced subgraphs is called $n$-universal for $X$. Moreover, we say that $U^{(n)}$ is a proper $n$-universal graph for $X$ if it belongs to $X$. In the present paper, we construct a proper $n$-universal graph for the class of split permutation graphs. Our solution includes two ingredients: a proper universal 321-avoiding permutation and a bijection between 321-avoiding permutations and symmetric split permutation graphs. The $n$-universal split permutation graph constructed in this paper has $4n^3$ vertices, which means that this construction is order-optimal.

math.CO↗

arXiv2013

An involution on bicubic maps and β(0,1)-trees

Bicubic maps are in bijection with β(0,1)-trees. We introduce two new ways of decomposing β(0,1)-trees. Using this we define an endofunction on β(0,1)-trees, and thus also on bicubic maps. We show that this endofunction is in fact an involution. As a consequence we are able to prove some surprising results regarding the joint equidistribution of certain pairs of statistics on trees and maps. Finally, we conjecture the number of fixed points of the involution.

math.CO↗

arXiv2013

The 1-box pattern on pattern avoiding permutations

This paper is continuation of the study of the 1-box pattern in permutations introduced by the authors in \cite{kitrem4}. We derive a two-variable generating function for the distribution of this pattern on 132-avoiding permutations, and then study some of its coefficients providing a link to the Fibonacci numbers. We also find the number of separable permutations with two and three occurrences of the 1-box pattern.

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