Word-representability of Toeplitz graphs
Gi-Sang Cheon, Jinha Kim, Minki Kim, Sergey Kitaev

TL;DR
This paper explores the word-representability of Toeplitz graphs, a class of Riordan graphs, establishing conditions for their representability, constructing examples, and pioneering the study of infinite word-representable graphs.
Contribution
It introduces the concept of word-representability for Toeplitz graphs, merges Riordan matrix theory with graph representation, and initiates the study of infinite word-representable graphs.
Findings
Several classes of Toeplitz graphs are proven to be word-representable.
A method for constructing non-word-representable Toeplitz graphs is provided.
First examples and discussion of infinite word-representable graphs are presented.
Abstract
Distinct letters and alternate in a word if after deleting in all letters but the copies of and we either obtain a word of the form (of even or odd length) or a word of the form (of even or odd length). A graph is word-representable if there exists a word over the alphabet such that letters and alternate in if and only if is an edge in . In this paper we initiate the study of word-representable Toeplitz graphs, which are Riordan graphs of the Appell type. We prove that several general classes of Toeplitz graphs are word-representable, and we also provide a way to construct non-word-representable Toeplitz graphs. Our work not only merges the theories of Riordan matrices and word-representable graphs via the notion of a Riordan graph, but also it provides the first systematic study of…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Geometric and Algebraic Topology · Advanced Graph Theory Research
Word-representability of Toeplitz graphs
Gi-Sang Cheon Applied Algebra and Optimization Research Center, Department of Mathematics, Sungkyunkwan University, Suwon, Republic of Korea. [email protected]
Minki Kim Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea. [email protected]
Jinha Kim Department of Mathematical Sciences, Seoul National University, Seoul, Republic of Korea. [email protected]
Sergey Kitaev Department of Computer and Information Sciences, University of Strathclyde, Glasgow, United Kingdom. [email protected]
Abstract
Distinct letters and alternate in a word if after deleting in all letters but the copies of and we either obtain a word of the form (of even or odd length) or a word of the form (of even or odd length). A graph is word-representable if there exists a word over the alphabet such that letters and alternate in if and only if is an edge in .
In this paper we initiate the study of word-representable Toeplitz graphs, which are Riordan graphs of the Appell type. We prove that several general classes of Toeplitz graphs are word-representable, and we also provide a way to construct non-word-representable Toeplitz graphs. Our work not only merges the theories of Riordan matrices and word-representable graphs via the notion of a Riordan graph, but also it provides the first systematic study of word-representability of graphs defined via patterns in adjacency matrices. Moreover, our paper introduces the notion of an infinite word-representable Riordan graph and gives several general examples of such graphs. It is the first time in the literature when the word-representability of infinite graphs is discussed.
Keywords: Toeplitz graph; word-representable graph; Riordan graph; pattern
AMS classification: 05C62, 15B05, 68R15
1 Introduction
In this paper, we merge the theories of Riordan matrices and word-representable graphs via the notion of a Riordan graph introduced recently in [6]. More precisely, we focus on the studies of Riordan graphs of the Appell type, which are known in the literature as Toeplitz graphs. We give various (general) conditions on (non-)word-representability of Toeplitz graphs leaving a complete classification in this research direction as a grand open question.
In this paper, for a word (or letter) , the word is denoted by . We also let denote the set .
1.1 Riordan matrices
For any integral domain , we consider the ring of formal power series
[TABLE]
If there exists a pair of generating functions with such that for each integer , then the matrix is called a Riordan matrix, or Riordan array, generated by and . Usually, we write . Since , every Riordan matrix is an infinite lower triangular matrix. In particular, if a Riordan matrix is invertible then it is a proper Riordan matrix. Note that is invertible if and only if , and . Some well known Riordan matrices are as follows.
The Pascal triangle
[TABLE]
The Catalan triangle
[TABLE]
The Fibonacci matrix
[TABLE]
For a Riordan matrix over , the -matrix defined by
[TABLE]
is called a binary Riordan matrix, and it is denoted by . The leading principal matrix of order in (resp., ) is denoted by (resp., ).
1.2 Riordan graphs
The notion of a Riordan graph was introduced in [6]. A simple labelled graph with vertices is a Riordan graph of order if the adjacency matrix of is an symmetric -matrix given by
[TABLE]
for some Riordan matrix over . We denote such by , or simply by when the matrix is understood from the context, or it is not important. A simple unlabelled graph is a Riordan graph if at least one of its labelled copies is a Riordan graph. However, only labelled graphs are of interest to us in this paper.
So, for a Riordan graph on vertices defined by , the adjacency matrix satisfies the following:
- •
its main diagonal entries are all 0, and
- •
its lower triangular part below the main diagonal is the binary Riordan matrix .
Note that the infinite graph
[TABLE]
is well defined, and we call it the infinite Riordan graph corresponding to the pair .
For example, the Riordan graph in Figure 1 is the complete graph , while is the infinite complete graph. Another example of a Riordan graph is the Catalan graph of order , which is defined as ; see Figure 2 for the case of .
1.3 Toeplitz graphs
A Riordan graph with is called a Riordan graph of the Appell type. For example, the Fibonacci graph is of such a type. The class of Riordan graphs of the Appell type is also known as the class of Toeplitz graphs. Originally, a Toeplitz graph is defined as a graph with and
[TABLE]
See [8] and references therein for examples of results in the literature on Toeplitz graphs.
Throughout this paper, we denote by the Toeplitz graph on vertices which is defined by
[TABLE]
where . For instance, the Fibonacci graph can be written as , or since .
1.4 Word-representable graphs
Suppose that is a word over some alphabet and and are two distinct letters in . We say that and alternate in if after deleting in all letters but the copies of and we either obtain a word of the form (of even or odd length) or a word of the form (of even or odd length).
A graph is word-representable if there exists a word over the alphabet such that letters and alternate in if and only if is an edge in . Such a word is called ’s word-representant. In this paper we assume to be for some . For example, the cycle graph on 4 vertices labeled by 1, 2, 3 and 4 in clockwise direction can be represented by the word 14213243. Note that a complete graph can be represented by any permutation of , while an edgeless graph (i.e. empty graph) on vertices can be represented by .
There is a long line of research on word-representable graphs, which is summarised in the recently published book [13] and the survey paper [12]. The roots of the theory of word-representable graphs are in the study of the celebrated Perkins semigroup [15, 18] which has played a central role in semigroup theory since 1960, particularly as a source of examples and counterexamples. However, the most interesting aspect of word-representable graphs from an algebraic point of view seems to be the notion of a semi-transitive orientation [11], which generalizes partial orders. It was shown in [11] that a graph is word-representable if and only if it admits a semi-transitive orientation (see Section 4 for a definition of a semi-transitive orientation).
More motivation points to study word-representable graphs include the fact exposed in [13] that these graphs generalize several important classes of graphs such as circle graphs [5], 3-colourable graphs [1] and comparability graphs [17]. Relevance of word-representable graphs to scheduling problems was explained in [11] and it was based on [10]. Furthermore, the study of word-representable graphs is interesting from an algorithmic point of view as explained in [13]. For example, the Maximum Clique problem is polynomially solvable on word-representable graphs [13] while this problem is generally NP-complete [3]. Finally, word-representable graphs are an important class among other graph classes considered in the literature that are defined using words. Examples of other such classes of graphs are polygon-circle graphs [16] and word-digraphs [2].
In relation to the main focus in our paper, one can prove the following theorem.
Theorem 1.1**.**
Toeplitz graphs and , , are word-representable for any .
Proof.
It is easy to see by induction on the number of vertices that , and thus , is 3-colorable. Indeed, the base case is trivial, and assuming is 3-colorable, we can obtain by increasing the labels of its vertices by 1, and adding the new vertex labeled by 1 to it. Since the new vertex is connected to (no more than) two vertices, there is at least one colour of three colours available for it. Since 3-colorable graphs are word-representable [11], we are done. ∎
1.5 Infinite word-representable graphs
For an infinite graph defined by the relation (1) we say that is word-representable if each finite graph is word-representable. We note that the notion of an infinite word-representable graph was never considered in the literature.
Define the index of word-representability IWR() of an infinite graph as the largest such that is word-representable. Since any graph on at most five vertices is word-representable [13], we have that IWR() for any . If is word-representable, we let IWR().
As corollaries to Theorems 1.1, 2.1 and 2.4 and Corollary 2.3 and 1.3, we obtain many examples of infinite word-representable graphs. In particular, it follows from Corollary 2.3 that the Fibonacci matrix defines a Toeplitz graph with the index of word-representability . On the other hand, it can be checked using [9] that the Pascal triangle and the Catalan triangle define Riordan graphs with the index of word-representability 11 and 12, respectively. The smallest non-word-representable Pascal and Catalan graphs are given, respectively, by the following adjacency matrices
[TABLE]
Also, from Section 3 we see that for ,
[TABLE]
which are the smallest non-word-representable Riordan graphs.
1.6 Comparability graphs
An orientation of a graph is transitive, if the presence of the edges and implies the presence of the edge . An oriented graph is a comparability graph if admits a transitive orientation. A graph is permutationally representable if it can be represented by concatenation of permutations of (all) vertices. Thus, permutationally representable graphs are a subclass of word-representable graphs. The following theorem classifies these graphs.
Theorem 1.2** ([15]).**
A graph is permutationally representable if and only if it is a comparability graph.
Note that is a comparability graph if its adjacency matrix satisfies the following: any time when and for we also have . To see this, one can obtain a transitive orientation of by orienting each edge as whenever . In particular, we have the following statement by the transitivity of the congruence relation. Observe that in the Toeplitz graph a vertex is adjacent to a vertex if and only if .
Corollary 1.3**.**
For any positive integers and , the Toeplitz graph is permutationally representable.
1.7 Our results in this paper
We already stated some results on word-representability of Toeplitz graphs in Theorem 1.1 and Corollary 1.3. For another result, we note that the Toeplitz graph defined by is word-representable, because is a complete graph on vertices and it can be represented by any permutation of length . For word-representation of Riordan graphs beyond Toeplitz graphs, we note that the graph defined by , , is also word-representable because each of such graphs is clearly a tree, and any tree can be represented using two copies of each letter [13]. (Word-representability of a tree also follows from Theorem 1.2 since any tree is a comparability graph.) The observation on can be generalized to Riordan graphs defined by where is any polynomial of degree and . Indeed, any such Riordan graph is a forest, so that it is a comparability graph and is word-representable by Theorem 1.2.
In either case, the main results in this paper are establishing word-representability of
- •
for any positive integers such that in Theorem 2.1; and
- •
and for any positive integers and in Corollary 2.3, which generalizes Corollary 1.3; and
- •
for any positive integers satisfying either or in Theorem 2.4.
We will also show in Theorem 3.2 that word-representability of a graph implies that for each positive divisor of , is also word-representable. The latter gives a way to construct non-word-representable Toeplitz graphs.
1.8 Proofs in this paper
All of our general statements contain rigorous proofs, e.g. in terms of explicit words representing various graphs. However, in many other situations we have to refer to the results obtained using the freely-available user-friendly software [9] created by Marc Glen to keep the paper being of reasonable size. Each of such results can be verified by hand as follows. If we claim that a (small) graph is word-representable, then [9] can produce a word-representant for the graph, which can be used as a certificate. On the other hand, if we claim that a (small) graph is non-word-representable (based on the results of [9]), then this can be checked using the notion of a semi-transitive orientation (see [7] or [13, Section 4.5] for a detailed explanation, illustrated on a particular graph, of how to do such a check; also, see Section 4).
2 Word-representable Toeplitz graphs
In this section, we investigate for which the graph is word-representable for every . We support our claims by explicit constructions of word-representants. Given a subset of , we let (resp., ) denote the words obtained by arranging the elements of in the increasing (resp., decreasing) order.
2.1 Word-representability of
In this section, we prove that for every positive integer , the graph is word-representable for each non-negative integers and such that . A direct consequence implies that for every positive integers and the graph of the form either or is word-representable. Note that the special case of and follows from Corollary 1.3.
Theorem 2.1**.**
For any non-negative integers and such that , the graph is word-representable for any positive integer .
Proof.
If , then the graph is empty, which is word-representable. Thus we may assume that . If , then the graph is a complete graph, which is word-representable. Hence we may further assume that at least one of the integers and is positive.
To construct the word that represents the graph , we partition the set into , where for each ,
[TABLE]
We then define by the -uniform word (permutation)
[TABLE]
for , and by the -uniform word
[TABLE]
for . Similarly, we define by the -uniform word
[TABLE]
for , and by the -uniform word
[TABLE]
for . For , we define to be the empty word for any . Note that the empty word does not affect alternation of any pair of vertices, so we do not have to consider the case of separately. Now we claim that the graph can be represented by the word
[TABLE]
We refer the Reader to Example 2.2 right after the proof illustrating the construction of in the case of , , and .
Before we go through the details, we will briefly describe our proof idea. We start with the -uniform word which represents the complete graph of order . Let and be two vertices in the graph . Without loss of generality, we may assume that . Note that the word contains either or as an induced subword.
We will first show that if the vertices and are adjacent in the graph , then and alternate in each of the words and , in the same order as in the word . For instance, if and are adjacent in the graph and contains as an induced subword, then each of and contains either or as an induced subword.
Then we will show that if the vertices and are not adjacent in the graph , then and do not alternate in the word . Here, the words and play an important role. For instance, depending on the condition of and , the order of and in the subword of may be different from that in the word , breaking alternation of and in the word .
Case 1. Let for some . In this case, and are adjacent in , so and must alternate in . We consider two subcases.
(1) First we assume that . Then and the word contains as an induced subword. Thus it is sufficient to show that for each , both and contain either or as an induced subword.
For , we claim that contains as an induced subword if and contains as an induced subword if . For , and cannot appear in the word at the same time because . Thus it is clear that the -uniform word
[TABLE]
contains as an induced subword. For , if and do not appear in the word at the same time, then it is obvious that the -uniform word
[TABLE]
contains as an induced subword. If the word contains both and , then we have and . This implies that the -uniform word contains as an induced subword. Therefore, for each , contains either or as an induced subword.
For , we claim that contains as an induced subword if , and contains as an induced subword if . For , it is obvious that the -uniform word
[TABLE]
contains as an induced subword if and do not appear in the word at the same time. On the other hand, since we have , we observe that it is impossible to have both and . Hence if the word contains both and , then it must be that either or . In any case, the -uniform word contains as an induced subword. For , clearly the -uniform word
[TABLE]
contains as an induced subword since .
(2) Now we assume that . In this case, and the word contains as an induced subword. We will show that for each , and contain either or as an induced subword.
For , from the assumption , it is clear that the -uniform word
[TABLE]
contains as an induced subword when . Hence it remains to show that for , the -uniform word
[TABLE]
contains as an induced subword. This claim is obviously true when and do not appear in the word at the same time. If the word contains both and , then it must be that either or . For this, observe that it is impossible to have both and . If not, we have , which implies that . This contradicts to the fact that . Thus it follows that the -uniform word contains as an induced subword.
Finally, we claim that contains as an induced subword if , and contains as an induced subword if . For , as before, the statement is obviously true if and do not appear at the same time in the word . Note that from , we have that . Thus if the word contains both and , then it must be that and . Thus the -uniform word
[TABLE]
contains as an induced subword. For , since , the word cannot contain both and , so that the -uniform word
[TABLE]
contains as an induced subword.
Case 2. Let for some . In this case, and are not adjacent in , so and must not alternate in . We consider two subcases.
(1) First assume that . Then and the word contains as an induced subword. If , then we have , and hence the word contains both and . Thus the -uniform word
[TABLE]
contains as an induced subword, which implies that and do not alternate in . If , then since , the word contains exactly one and exactly one . Thus the -uniform word
[TABLE]
contains as an induced subword, so and do not alternate in .
(2) Next we assume that . Then and the word contains as an induced subword. Since , we have . Thus the word contains exactly one and exactly one . Therefore the word
[TABLE]
contains as an induced subword, so and do not alternate in .
Case 3. Let for some . Again, and are not adjacent in in this case, and hence and must not alternate in . The argument for this case is similar to that for Case 2, but here we will use the word instead of the word in Case 2.
(1) First assume that . Then and contains as an induced subword. Since , we have . Thus the word contains exactly one and exactly one . Thus the -uniform word
[TABLE]
contains as an induced subword, and hence and do not alternate in .
(2) Next we assume . Then and contains as an induced subword. If , then since , the word contains exactly one and exactly one . Thus the -uniform word
[TABLE]
contains as an induced subword. If , then the word contains exactly one and exactly one since . Thus the word
[TABLE]
contains as an induced subword. In either case, and do not alternate in . ∎
Example 2.2**.**
We illustrate the construction of the word in Theorem 2.1 in the case of , , and . In this case, , , , , , and . Thus, is obtained by concatenating the following words
[TABLE]
It can be checked using [9] that this indeed represents .**
Observe that in Theorem 2.1, we do not have to consider Case 2 if , and we do not have to consider Case 3 if . This allows us to provide shorter words that represent the graphs and , respectively. They will be described in the following corollary.
Corollary 2.3**.**
For any positive integers and , the graphs and are word-representable.
Proof.
Let and be fixed. Then the graph is precisely the case when and are positive and in Theorem 2.1. By the above observation, Case 1 and Case 2 in Theorem 2.1 imply that the graph can be represented by the word
[TABLE]
Now let and be fixed. Then the graph is precisely the case when and are positive and in Theorem 2.1. By the above observation, Case 1 and Case 3 in Theorem 2.1 imply that the graph can be represented by the word
[TABLE]
∎
2.2 Word-representability of
Here we deal with word-representability of under given assumptions. The condition that either or cannot be removed because of the existence of non-word-representable graphs examined by [9]. When we have and , the graph is not word-representable. Note that in this case we have . Moreover, the assertion of Theorem 2.4 is not naturally extended to arbitrary graphs of type . For instance, is not word-representable.
Theorem 2.4**.**
If positive integers such that satisfy or (when is even), is word-representable for any .
Proof.
Let be a partition of such that for each , .
Case 1. We first consider when . Note that are all distinct. If not, then it must be that for some , implying that can be divided by . Since , we can conclude that can be divided by , which is not true because .
Now for each , we define a -uniform word by
[TABLE]
and we define a -uniform word by
[TABLE]
We claim that the word represents the graph . Suppose and for some . Without loss of generality, we may assume that .
(1) Assume that either or and . Then and are not adjacent in , thus and should not alternate in . If , then the word contains as an induced subword, and the word contains as an induced subword. If we have and , then the word contains as an induced subword. In both cases, and do not alternate in the word .
(2) Otherwise, and are adjacent in , and hence and should alternate in . If , then for each , the word contains as an induced subword since , and the word contains as an induced subword. If , then for each , the word contains as an induced subword, and the word contains as an induced subword since and do not appear in the word at the same time. In both cases, and alternate in .
Case 2. Now we consider when is even and . In this case, the -uniform word
[TABLE]
represents the graph . To see this, suppose and for some , and assume that .
If , then and are not adjacent in . Observe that the word contains as an induced subword if , and that contains as an induced subword if . Thus and do not alternate in .
Otherwise, i.e. if , and are adjacent in . Clearly, the word contains as an induced subword if and contains as an induced subword if , where and and . Thus and alternate in . ∎
3 Non-word-representable Toeplitz graphs
As is mentioned above, not all Toeplitz graphs are word-representable. Using [9] we see that the smallest non-word-representable Toeplitz graph has nine vertices. An example of such a graph is given by the adjacency matrix
[TABLE]
In this section, we give a necessary condition on the word-representability of a Toeplitz graph. To do this, we first prove that the induced subgraph of a Toeplitz graph on the vertex subset is also a Toeplitz graph. The specific case when implies Theorem 3.12 (iii) in [6].
Lemma 3.1**.**
Let be an Riordan matrix defined by . For a positive integer , let and be the submatrix of induced by columns in . Then the matrix is a Riordan matrix defined by .
Proof.
For each such that , the entry of is equal to the entry of , which is equal to . Thus is the Riordan matrix which is defined by . ∎
Theorem 3.2**.**
Let be a word-representable Toeplitz graph. Then, for each positive divisor of , is word-representable.
Proof.
Let be the adjacency matrix of . Then is the Riordan matrix defined by
[TABLE]
In other words, is the Riordan matrix defined by where is given by for each . Then, by Lemma 3.1 we obtain that the submatrix of is the Riordan matrix defined by
[TABLE]
for . Thus is the adjacency matrix of the Toeplitz graph with vertices defined by
[TABLE]
Therefore, is an induced subgraph of , which implies that is also word-representable by the heredity of the word-representablity. ∎
Theorem 3.2 says that if a Toeplitz graph is not word-representable for some divisor of then the graph is not word-representable. This gives a way to construct non-word-representable Toeplitz graphs. For example, is not word-representable because is not word-representable. More generally, is not word-representable where ’s are words over of length since , so that Theorem 3.2 guarantees that among Toeplitz graphs on vertices, there are at least non-word-representable graphs.
4 Concluding remarks
In this paper we give several general classes of word-representable Toeplitz graphs, using explicit representation as a key approach. We were not able to apply similar approach to the other classes of Toeplitz graphs. In any case, further advances in the area, hopefully leading to a complete classification of word-representable Toeplitz graphs, or more generally Riordan graphs, may require usage of other tools, such as the powerful notion of a semi-transitive orientation. This notion has been used successfully in many situations (see [12] for an overview), because it allows to bypass dealing with complicated constructions on words, and we complete this paper with describing it.
The notion of a semi-transitive orientation was introduced in [11], but we follow [13, Section 4.1] to introduce it here. A graph is semi-transitive if it admits an acyclic orientation such that for any directed path with for all , , either
- •
there is no edge , or
- •
the edge is present and there are edges for all . In other words, in this case, the (acyclic) subgraph induced by the vertices is transitive (with the unique source and the unique sink ).
We call such an orientation semi-transitive. In fact, the notion of a semi-transitive orientation is defined in [11] in terms of shortcuts as follows. A semi-cycle is the directed acyclic graph obtained by reversing the direction of one edge of a directed cycle in which the directions form a directed path. An acyclic digraph is a shortcut if it is induced by the vertices of a semi-cycle and contains a pair of non-adjacent vertices. Thus, a digraph on the vertex set is a shortcut if it contains a directed path , the edge , and it is missing an edge for some ; in particular, we must have , so that any shortcut is on at least four vertices. Clearly, this definition is just another way to introduce the notion of a semi-transitive orientation presented above.
It is not difficult to see that all transitive (that is, comparability) graphs are semi-transitive, and thus semi-transitive orientations are a generalization of transitive orientations. A key theorem in the theory of word-representable graphs is presented next, and we expect it to be of great use in the study of word-representable Riordan graphs.
Theorem 4.1** ([11]).**
A graph is word-representable if and only if it admits a semi-transitive orientation (that is, if and only if is semi-transitive).
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (2016R1A5A1008055) and the Ministry of Education of Korea (NRF-2016R1A6A3A11930452).
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