1. Introduction
An important topic, on the boundary of graph theory and computer science, is finding efficient ways of encoding graphs. One possible method is to encode graphs as strings. Of course there are many potential methods to do this, but a natural way is to consider strings whose alphabet is the vertex set and the (non-)adjacencies are encoded based on properties of the strings.
One fairly natural way to do this, first formally studied by Kitaev and Pyatkin [Reference Kitaev and Pyatkin8] and motivated by earlier work of Kitaev and Seif from [Reference Kitaev and Seif9], is to encode edges between two vertices 
$x$ and 
$y$ by (strict) alternation of the characters 
$x$ and 
$y$ in the word. This gives rise to the word-representation of a graph. Unfortunately, not all graphs can be represented in this way, but the class of word-representable graphs has been well-studied in the literature.
Motivated by work on permutation patterns, a more general class of representations are the 
$k$-
$11$-representations. These representation were first suggested by Remmel, but first studied in detail by Cheon et al. in [Reference Cheon, Kim, Kim, Kitaev and Pyatkin2] with follow-up work in several papers, including [Reference Futorny, Kitaev and Pyatkin4] and the very recent [Reference Alshammari, Kitaev, Tang, Tao and Zhang1]. A more formal definition is given below, but here adjacency is again encoded by alternation, but up to 
$k$ violations of being strict are allowed. The 
$0$-
$11$-representable graphs, then, are word-representable graphs and it was shown in [Reference Cheon, Kim, Kim, Kitaev and Pyatkin2] that all graphs are 
$2$-
$11$-representable. The question of whether all graphs are 
$1$-
$11$-representable was left as an open question. Indeed in the recent [Reference Alshammari, Kitaev, Tang, Tao and Zhang1] this problem was settled for graphs on at most 
$8$ vertices only after some effort.
In this paper, we settle the question affirmatively, for all 
$n$, in a strong fashion. Our main results are as follows:
• In Theorem 2.2, we prove that every graph is permutationally
$1$-
$11$-representable, that is, it can be
$1$-
$11$-represented by a word which is the concatenation of permutations of vertices. Our proof, which is constructive, also gives a variety of extensions to more restrictive representations and also to orientations of graphs. Not only does this answer the open question in [Reference Cheon, Kim, Kim, Kitaev and Pyatkin2] of the
$1$-
$11$-representability of all graphs, but the (stronger) question asked in [Reference Cheon, Kim, Kim, Kitaev and Pyatkin2] about the existence of permutational representations.• In Theorem 2.7, we prove the impossibility of even–odd-representations for almost all graphs – that is, representations by words where the parity and not the number of occurences of the
$11$ pattern determine adjacency. This answers a question raised by Ian Wanless in a 2018 conferenceFootnote 1 after a talk on the existence of
$2$-
$11$-representations. We completely characterize the graphs that are permutationally representable in such a way (see Theorem 2.8.)• We further study the length of these representations. We prove as Theorem 3.6 that every graph has a
$1$-
$11$-representation consisting of the concatenation of
$O(n)$ permutations, while proving that some graphs require
$\Omega(n/\log n)$ permutations to permutationally represent, so our bounds are tight within a factor of
$\log(n)$.• We, in fact, present a slightly stronger lower bound (Theorem 3.1) showing that there are graphs that require
$\Omega(n^{2}/\log n)$ characters in any
$1$-
$11$-representation – whether permutational or not.• Related to this lower bound: it is natural to suspect that graphs which are hard to word-represent are hard to
$1$-
$11$-represent. We show this is not always the case. We show (see Theorem 3.4) that the crown graph
$H_{n,n}$ – obtained by deleting a perfect matching from
$K_{n,n}$ – is permutationally
$1$-
$11$-representable by
$4$ permutations, independent of
$n$, despite the fact that it requires
$\lceil n/2\rceil$ copies of each letter in any word-representant.• Our construction (i.e. Theorem 3.6) easily adapts to give permutational
$k$-
$11$-representations for any
$k\geq 1$ using
$O(n)$ permutations. The previous construction from [Reference Cheon, Kim, Kim, Kitaev and Pyatkin2] building
$2$-
$11$-representations for arbitrary graphs uses
$\Omega(n^2)$ permutations, so our construction yields a significant improvement. The lower bound we present (Theorem 3.1) also adapts, so this is also tight within a logarithmic factor.
As a reminder of asymptotic notation, for functions 
$f(n)$ and 
$g(n)$, we say 
$f(n) = O(g(n))$ if there exists a constant 
$C \gt 0$ so that 
$\lvert f(n)\rvert \leq C \lvert g(n) \rvert$ for 
$n$ sufficiently large. Meanwhile, 
$f(n) = o(g(n))$ means that 
$f(n)/g(n) \to 0$ as 
$n \to \infty$; in particular if 
$f(n) = o(1)$, then 
$f(n) \to 0$. A function 
$f(n) = \Omega(g(n))$ if 
$g(n) = O(f(n))$. All logarithms in the paper are taken to be the base-2 logarithm, unless otherwise specified. (In this paper, the base of the logarithm seldom matters as the logarithms are within 
$O( \cdot)$ notation, and changing the base only changes the implied constant.)
The remainder of the paper is organized as follows: in 
$\S$ 2, we give a simple construction which can be used to construct a 
$1$-
$11$-representation of all graphs, and discuss some extensions to more general representations and also to representing acyclic orientations of graphs. We further consider the existence of even–odd representations of graphs in this section. Then in 
$\S$ 3, we study the lengths of representations, both presenting our lower bounds and giving the optimizations to the main construction that lead to improved upper bounds.
2.
$1$-
$11$-representations and even–odd-representations of graphs
Let 
$k\geq 0$. A graph 
$G=(V,E)$ is 
$k$-
$11$-representable if there is a word 
$w$ on alphabet 
$V$ such that 
$u\sim v$ if and only if the number of occurrences of factors of the restriction 
$w|_{uv}$ that are 
$uu$ or 
$vv$ is less than or equal to 
$k$. We call the existence of such a factor a defect. A graph is permutationally 
$k$-
$11$-representable if it is 
$k$-
$11$-representable by a word that is a concatenation of permutations of the vertex set.
Before we begin with the thrust of the section – showing that every graph is permutationally 
$1$-
$11$-representable – we quickly present the following helpful observation.
Lemma 2.1. Let 
$w$ be a word 
$1$-
$11$-representing a graph 
$G = (V,E)$ and let 
$\pi$ be the permutation of 
$V$ consisting of the final occurrences of each letter in 
$w$. Then 
$w\pi$ 
$1$-
$11$-represents 
$G$.
Proof. Let 
$u,v \in V$. If 
$u\sim v$ then 
$u$ and 
$v$ have at most one defect in 
$w$, and appending 
$\pi$ adds alternation rather than a defect. If 
$u \not\sim v$ then in 
$w$, 
$u$ and 
$v$ have more than one defect in 
$w$ and thus in 
$w\pi$ as well. Thus appending 
$\pi$ preserves both edges and non-edges.
2.1. Constructing
$1$-
$11$-representations
We begin with the main construction showing all graphs have 
$1$-
$11$-representations. Though there are optimizations that can be done, we present it here in such a way to easily admit generalizations that we then consider.
Theorem 2.2 Let 
$G = (V,E)$ be a graph. Then there is a word 
$w$ over alphabet 
$V$ permutationally 
$1$-
$11$-representing 
$G$.
Proof. Order the vertices of 
$V$ arbitrarily as 
$v_1, \dots, v_{n}$. We prove that the induced subgraph on 
$v_1, \dots, v_t$ is permutationally 
$1$-
$11$-representable by induction on 
$t$. For the base case 
$t=1$, the word 
$v_1$ is a permutational 
$1$-
$11$-representation of the induced subgraph on 
$v_1$.
Now suppose 
$t \geq 2$.
Let 
$v=v_t$. By the inductive hypothesis there exists a word 
$w'$ over alphabet 
$\{v_1, \dots, v_{t-1}\}$ permutationally 
$1$-
$11$-representing the induced subgraph on 
$\{v_1, \dots, v_{t-1}\}$: 
$w' = \pi_1\pi_2\cdots\pi_k$. Write the order of the vertices in the final permutation as 
$\pi_k=x_1x_2\cdots x_{t-1}$; note that 
$x_1, x_2, \dots, x_{t-1}$ is a permutation of 
$v_1, \dots, v_{t-1}$.
For 
$1 \leq i \leq t-1$ we build a concatenation of 
$1$ or 
$3$ permutations based on 
$x_i$. Let
\begin{equation*}
\sigma_i = \left\{\begin{array}{l@{\qquad}l}
x_1x_2\cdots x_{i-1}vx_i\cdots x_{t-1} & \text{if $x_i \sim v$}\\[3pt]
(x_1x_2\cdots x_{i-1}vx_i\cdots x_{t-1})(x_1x_2\cdots x_{i}vx_{i+1}\cdots x_{t-1})\\
\quad \times\,(x_1x_2\cdots x_{i-1}vx_i\cdots x_{t-1}) & \text{if $x_i \not\sim v$}
\end{array}\right.\!.
\end{equation*}Now consider the word
\begin{equation*}
w = \pi_1v\pi_2v\cdots \pi_kv\sigma_{t-1}\sigma_{t-2}\cdots \sigma_1.
\end{equation*} By the construction of 
$w$ and Lemma 2.1, the subword of 
$w$ induced by 
$X =\{x_i : 1\leq i\leq t-1\}$ 
$1$-
$11$-represents 
$G[X]$. Thus we need only to check edges and non-edges on 
$v$.
• If
$x_i \sim v$, then in
$\pi_1v\pi_2v\cdots \pi_kv\sigma_{t-1}\sigma_{t-2}\cdots \sigma_{i+1}$ the pattern of
$x_i$ and
$v$ is
$x_ivx_iv\cdots x_iv$ and in
$\sigma_{i}\sigma_{i-1}\cdots \sigma_1$ the pattern is
$vx_ivx_i\cdots vx_i$. Together the pattern of
$x_i$ and
$v$ is only 1 defect away from alternation.• If
$x_i \not\sim v$, then in
$w$ the pattern of
$x_i$ and
$v$ is
$x_ivx_i\dots vx_ix_ivvx_i\dots$, which actually has
$3$ defects from being alternating.
Therefore 
$w$ truly does 
$1$-
$11$-represent 
$G$.
This construction has a number of optimizations and generalizations. For instance, in the definition of 
$\sigma_i$ when 
$x_i \not\sim v$, one only really needs the first two permutations. Making this change will shorten the resulting representation, though the number of defects for non-adjacent vertices will no longer always be exactly three – the last non-adjacency will only have 
$2$. One can also process various bunches of vertices in bulk, depending on the adjacencies. These types of modifications, however, will still generate representations that are the concatenation of a quadratic number of permutations. In the next section we describe, in Theorem 3.6, a rather more complicated optimization which concatenates only a linear number of permutations to realize 
$G$. For that reason, we choose to keep this simple and easy to verify version here. In terms of generalizations, we mention a few.
One construction, extremely useful in the theory of word-representability, is that of an orientation from a word. Given a word representing a graph, one can build an acyclic orientation by ordering vertices according to their first appearance in the word; that is, an edge between 
$x$ and 
$y$ is oriented as 
$x \to y$ if the first occurrence of 
$x$ occurs before the first occurrence of 
$y$ in the word. In the case of standard word-representations (i.e. 
$0$-
$11$-representations), these orientations are ‘semi-transitive’, and indeed having a word-representation is equivalent to admitting a semi-transitive orientation as shown in [Reference Halldórsson, Kitaev and Pyatkin6].
In the case of 
$1$-
$11$-representations, this orientation still makes sense. That is, a word 
$1$-
$11$-representing a graph can be thought of as representing an acyclically oriented graph. One can recover almost immediately from our construction presented above the following strengthening.
Theorem 2.3 Let 
$G = (V,E)$ be an acyclically oriented graph. Then there is a word 
$w$ over alphabet 
$V$ permutationally 
$1$-
$11$-representing 
$G$ as an oriented graph.
Proof. The only modification necessary is to carefully choose the ordering 
$v_1, \dots, v_n$. One chooses 
$v_{n}$ to be a sink, which exists because the orientation is acyclic. Then one inductively chooses 
$v_{n-i}$ to be a sink in the graph induced on 
$V \setminus \{v_n, v_{n-1}, \dots, v_{n-i+1}\}$ for 
$i = 1, \dots, n$ – which again exists by the acyclicity of the orientation on the induced subgraph.
It is then easy to see that the construction described above will begin with the permutation 
$v_1v_2\dots v_{n}$ which forces the orientation to coincide with that on the graph.
Another generalization is to differ the pattern of defects occurring in words. Given integers 
$(k,\ell)$ (with 
$k\neq\ell$) we say that 
$G$ is 
$(k,\ell)$-
$11$-representable if there is a word 
$w$ so that:
• Whenever
$x \sim y$, the pattern
$xx$ or
$yy$ appears in the subword
$w|_{xy}$ exactly
$k$ times.• Whenever
$x \not\sim y$, the pattern
$xx$ or
$yy$ appears in the subword
$w|_{xy}$ exactly
$\ell$ times.
As observed in the proof, the 
$1$-
$11$ construction given in Theorem 2.2 actually yields a 
$(1,3)$-
$11$-representation of 
$G$. This is easily generalized to other values of 
$(k, \ell)$.
Theorem 2.4 Suppose 
$k$ and 
$\ell$ are distinct positive integers with the same parity. Let 
$G=(V,E)$ be a graph. Then there is a word 
$w$ over 
$V$ permutationally 
$(k, \ell)$-
$11$-representing 
$G$.
Proof. As noted above, the proof of Theorem 2.2 above gives a 
$(1,3)$-
$11$-representation. If 
$k$ and 
$\ell$ are both odd, then the modification of the proof is very straightforward: in the definition of the 
$\sigma_i$, instead of adding either 
$1$ or 
$3$ permutations, one adds 
$k$ or 
$\ell$ permutations – moving vertex 
$v$ back and forth past 
$x_i$. Each of these introduces exactly one 
$x_ix_i$ or one 
$vv$. Since 
$k$ and 
$\ell$ are odd, the last of these will always have 
$v$ to the left of 
$x_i$, which is what is necessary to continue the construction without creating any more defects when considering 
$x$ and 
$v_i$.
In the case where 
$k$ and 
$\ell$ are even (and hence, 
$k, \ell \geq 2$), one can easily create a 
$(k-1, \ell-1)$-
$11$-representation 
$w$ as already described. This can easily be turned into a 
$(k,\ell)$-
$11$-representation by taking the permutation 
$\pi$ consisting of the final occurrences of each letter in order, reversing 
$\pi$ to get a permutation 
$\sigma$, and then taking 
$w' = w\sigma.$ This reversal introduces exactly one new defect into each letter pair.
2.2. Even–odd-representations of graphs
The restriction in Theorem 2.4 that 
$k$ and 
$\ell$ have the same parity is perhaps curious. A related question, raised in 2018 by Ian Wanless after a talk on 
$2$-
$11$-representations of graphs, is the following: is it possible to represent every graph by a word so that for every edge the parity of the number of 
$11$ patterns is (say) even, and for every non-edge the number of 
$11$ patterns is odd? It turns out that the answer to this question is rather emphatically no: a very small fraction of graphs can be represented in this form, whether permutationally or not. This also shows that the restriction that 
$k$ and 
$\ell$ must have the same parity, above, is unavoidable.
To make this precise, we say that 
$G$ is even-odd-representable if there is a word 
$w$ whose alphabet is the vertex set so that:
• Whenever
$x \sim y$, the pattern
$xx$ or
$yy$ appears in the subword
$w|_{xy}$ an even number of times.• Whenever
$x \not\sim y$, the pattern
$xx$ or
$yy$ appears in the subword an odd number of times.
We begin by making an observation regarding the parity of occurrences of 
$11$-patterns in words.
Lemma 2.5. Suppose 
$w$ is a word in alphabet 
$\{x,y\}$. Let 
$n_x$ and 
$n_y$ be the number of occurrences of 
$x$ and 
$y$ in 
$w$ respectively, and let 
$n =n_x + n_y$ be the length of 
$w$. Further, let
\begin{equation*}
e(w) = \begin{cases} 1 & \mbox{if the first and last character of $w$ are the same (or $n$=1)}\\
0 & \mbox{else.}
\end{cases} \end{equation*} Let 
$N(w)$ denote the number of occurrences of the pattern 
$11$ in 
$w$. Then
\begin{equation*}
N(w) \equiv n +e(w) \equiv n_x + n_y + e(w) \pmod{2}.
\end{equation*}Proof. Removing 
$v_n$ decreases 
$n$ by 1. If 
$v_n =v_{n-1}$ then 
$e(w)$ remains the same, while 
$N(w)$ decreases by 1. Otherwise, 
$e(w)$ changes parity, while 
$N(w)$ remains the same. In either case, the parity of 
$n-e(w)-N(w)$ does not change.
The following observation then easily follows.
Lemma 2.6. Suppose 
$G$ is even–odd-representable. Then there is a word 
$w'$ representing 
$G$ where each vertex appears at most 
$3$ times.
Proof. Suppose 
$G$ is even–odd-representable and fix a word 
$w$ even–odd-representing 
$G$. Suppose some vertex 
$x$ appears at least 
$4$ times in 
$w$. Let 
$w'$ be the word obtained by removing two of the middle appearances of 
$x$ from 
$w$, leaving the rest unchanged. We claim 
$w'$ also represents 
$G$. Indeed, removing occurrences of 
$x$ can only affect incidences between 
$x$ and other vertices and, when considering the existence of an edge between 
$x$ and 
$v$ we observe that if 
$w|_{xv}$ is the string restricted to just 
$x$ and 
$v$
• The parity of the length of
$w|_{xv}$ and
$w'|_{xv}$ is the same, as
$w'|_{xv}$ is two characters shorter than
$w|_{xv}$.• The initial and final characters of
$w|_{xv}$ and
$w'|_{xv}$ are the same, as only the middle instances of
$x$ were changed.
But then, per Lemma 2.5, the parity of the number of occurrences of a 
$11$-pattern in 
$w|_{xv}$ and 
$w'|_{xv}$ are the same – so 
$w$ and 
$w'$ represent the same graph. The shortest word representing 
$G$ thus has the desired property.
Finally we are ready to prove
Theorem 2.7 Almost every graph is not even–odd-representable; that is, if 
$G$ is an 
$n$ vertex graph chosen uniformly at random, then with probability 
$1-o(1)$, 
$G$ is not even–odd-representable.
Proof. Per Lemma 2.6, every even–odd-representable graph has order a string representing it of length at most 
$3n$. Quite naively, there are at most 
$\sum_{t=1}^{3n} n^{t} = (1+o(1))n^{3n} = 2^{(1+o(1))3n\log(n)}$ such strings representing labelled graphs of order 
$n$, and hence at most that many even–odd-representable graphs. On the other hand, there are 
$2^{\binom{n}{2}} = 2^{(1+o(1))n^2/2}$ labelled graphs on 
$n$ vertices. Clearly
\begin{equation*}
\frac{2^{(1+o(1))3n\log(n)}}{2^{(1+o(1))n^2/2}} \to 0,
\end{equation*}and thus almost every graph is not even–odd-representable.
A simple extension to Lemma 2.6 allows us to completely characterize permutationally even–odd-representable graphs. To set it up, we quickly state the relevant definitions.
We recall that a (strict) partial order 
$(P, \prec)$ is a relation 
$\prec$ on 
$P$ that is irreflexive, anti-symmetric and transitive. A linear extension of 
$(P, \prec)$ is a total ordering consistent with the partial ordering. A realizer of 
$(P, \prec)$ is a collection of linear extensions 
$\pi_1, \dots, \pi_t$ so that 
$x \prec y$ in 
$(P,\prec)$ if and only if 
$x \prec y$ in all 
$\pi_{i}$. The dimension of a poset, then, is the minimum cardinality of a realizer. Partially ordered sets, their realizers, and their dimension are closely related to the word-representation of graphs; see eg. [Reference Hefty, Horn, Muir and Owens7]. The compatibility graph of a poset 
$(P, \prec)$ is a graph so on vertex set 
$P$, so that 
$a \sim b$ if and only if either 
$a \prec b$ or 
$b \prec a$.
Theorem 2.8 
$G$ is permutationally even–odd-representable if and only if it is the comparability graph of a poset of dimension at most two.
Proof. Suppose 
$G$ is permutationally even–odd-representable; let 
$w = \pi_1 \dots \pi_t$ be the word. We begin by noting that one may assume 
$t \leq 2$. Otherwise, consider 
$w'$ obtained by removing one of the middle permutations. Then for each pair 
$u,v$ over vertices, considering the restriction 
$w|_{uv}$ of 
$w$ to those vertices, we note:
• The parity of the length of
$w|_{uv}$ and
$w'|_{uv}$ is the same, as the number of
$u$s and
$v$s were both reduced by
$1$.• The initial and final characters of
$w|_{uv}$ and
$w'|_{uv}$ is the same, as only the middle instances of
$u$ and
$v$ were changed.
Thus 
$w$ and 
$w'$ even–odd-represent the same graph.
Taking 
$w$ to be the shortest word, we see that either 
$w = \pi_1$, in which case 
$G$ is complete (and the comparability graph of a poset of dimension 
$1$), or 
$w = \pi_1\pi_2$. In the latter case, it is easy to see that even–odd-representability is the same as word-representability and the graph obtained is the comparability graph of the poset with realizer 
$\pi_1, \pi_2$.
Conversely, if 
$G$ is the comparability graph of a poset of dimension at most two, the concatenation of a realizer viewed as a word gives a word-representation of its comparability graph – but word-representation is easily seen to correspond to even–odd-representation in this case.
As mentioned in the beginning of the section, this immediately implies the following corollary, as otherwise the words generated would contradict Theorem 2.8.
Corollary 2.9. In Theorem 2.4, the condition that 
$k$ and 
$\ell$ have the same parity is necessary.
3. The length of
$1$-
$11$-representations
In this section, we study the length of 
$1$-
$11$-representations of graphs. In Theorem 2.2, we showed that all graphs admit a permutational 
$1$-
$11$-representations. The simple construction given above uses 
$\Omega(n^{2})$ permutations to represent a graph. In this section, we prove lower bounds on the length of a 
$1$-
$11$-representation, and then show that the construction of 
$\S$ 2 can be optimized to use far fewer permutations (albeit at the cost of some complexity.)
For this section, we restrict ourselves to dealing with 
$1$-
$11$-representations, although simple modifications allow one to recover essentially the same asymptotic bounds for 
$k$-
$11$-representations for any 
$k \geq 1$.
To this end, we let
•
$\mathcal{R}_{\pi}(G)$ denote the permutational
$1$-
$11$-representation number of
$G$ – the minimum number of permutations in any
$1$-
$11$ permutational representation of
$G$. We further let•
$\mathcal{R}(G)$ denote the
$1$-
$11$-representation number of G, the minimum number of characters in any
$1$-
$11$-representation of
$G$.
Note that 
$\mathcal{R}(G) \leq n \mathcal{R}_{\pi}(G)$ as the concatenation of 
$\mathcal{R}_{\pi}(G)$ permutations of length 
$n$ is a word-representing 
$G$ on 
$n\mathcal{R}_{\pi}(G)$ characters.
3.1. Lower bounding representation length
We begin by offering lower bounds for the length of any 
$1$-
$11$-representation (or even 
$k$-
$11$- representation) of a graph. These follow by simple counting arguments; unfortunately it seems somewhat difficult to prove that any particular graph has a large representation number.
Theorem 3.1 There exist 
$n$ vertex graphs with
\begin{equation*}\mathcal{R}(G) \geq (1+o(1))\frac{n^2}{2\log(n)}.\end{equation*}Proof. Suppose 
$t$ is such that every labelled 
$n$ vertex graph has 
$\mathcal{R}(G) \leq t$. On one hand, there are at most 
$\sum_{k=1}^{t} n^{k} = O(n^{t}) = 2^{(1+o(1))t \log(n)}$ such words. On the other hand, there are 
$2^{\binom{n}{2}} = 2^{(1+o(1))n^{2}/2}$ labelled graphs on 
$n$ vertices. As each word represents a single graph, we must have
\begin{equation*}
2^{(1+o(1))t \log(n)} \geq 2^{\binom{n}{2}},
\end{equation*} and hence 
$t \geq (1+o(1)) \frac{n^2}{2\log n}$.
An immediate corollary is that
Corollary 3.2. There exist 
$n$ vertex graphs with
\begin{equation*}\mathcal{R}_{\pi}(G) \geq (1+o(1)) \frac{n}{2\log(n)}.\end{equation*} There are 
$2^{(1+o(1))n^{2}/4}$ labelled bipartite graphs on partite sets 
$(X,Y)$ with 
$\lvert X \rvert = \lfloor n/2 \rfloor$ and 
$\lvert Y \rvert = \lceil n/2 \rceil$, so the same argument gives
Corollary 3.3. There exist 
$n$-vertex bipartite graphs with
\begin{equation*}\mathcal{R}(G) \geq (1+o(1)) \frac{n^2}{4\log n},\end{equation*}and
\begin{equation*}\mathcal{R}_{\pi}(G) \geq (1+o(1)) \frac{n}{4\log n}.\end{equation*} Although these counting arguments are quite naive – and indeed imply that almost every 
$n$ vertex graph satisfy the existential lower bound of Theorem 3.1 – it seems difficult to give a non-trivial lower bound on 
$\mathcal{R}(G)$ for any particular graph. One potential idea is to look at graphs which are hard to word-represent efficiently. In [Reference Cheon, Kim, Kim, Kitaev and Pyatkin2], the authors showed that a word-representation of a graph can be turned into a 
$1$-
$11$-representation by appending at most 
$n$ characters.
The crown graph, 
$H_{n,n}$, is a bipartite graph on 
$2n$ vertices, where each part has size 
$n$. It consists of a complete bipartite graph 
$K_{n,n}$ with a matching removed. It is known, see [Reference Glen, Kitaev and Pyatkin5], that this has a large word-representation number and is perhaps a natural candidate for a graph with a large 
$1$-
$11$-representation number. Unfortunately, this turns out not to be the case.
Theorem 3.4 The crown graph 
$H_{n,n}$ satisfies
\begin{equation*}\mathcal{R}_{\pi}(H_{n,n}) \leq 4.\end{equation*}Proof. Denote the partite sets of 
$H_{n,n}$ by 
$A = \{a_i : 1\leq i\leq n\}$ and 
$B = \{b_i : 1\leq i \leq n\}$ where 
$a_i \sim b_j$ if 
$i \neq j$. The word
\begin{equation*}
(a_1a_2 \cdots a_n b_1b_2 \cdots b_n)(a_na_{n-1}\cdots a_1b_nb_{n-1}\cdots b_1)(b_1a_1b_2a_2\cdots b_na_n)(a_1b_1a_2b_2 \cdots a_nb_n),
\end{equation*} 
$1$-
$11$-represents 
$H_{n,n}$.
We can, however, prove one general lower bound which can be applied to specific graphs inspired by a previous result of the authors and Owens from [Reference Hefty, Horn, Muir and Owens7] on word-representations. To this end, we make a definition. Let 
$N(x)$ denote the neighbourhood of a vertex 
$x$ in a graph. For a graph 
$G$ and 
$X \subseteq V(G)$, we define
\begin{equation*}
\mathcal{N}(X) = \{N(z) \cap X: z \in (V(G) \setminus X)\}.
\end{equation*} 
$\lvert \mathcal{N}(X) \rvert$ then is the number of distinct neighbourhoods that other vertices have within 
$X$. Then
Theorem 3.5 Suppose 
$G$ is a graph, and 
$X \subseteq V(G)$. Then
\begin{equation*}
\mathcal{R}_{\pi}(G) \geq \frac{\log \lvert\mathcal{N}(X)\rvert}{\log (\lvert X \rvert+1)}.
\end{equation*}Proof. Fix a permutational representation of 
$G$ with 
$t = \mathcal{R}_{\pi}(G)$ permutations. Fix a vertex 
$z$. Within each of these 
$t$ permutations, 
$z$ sits in one of 
$\lvert X \rvert+1$ locations with respect to the vertices of 
$X$ – before the first, between the first and second, 
$\ldots$, or after the last. These locations, in the 
$t$ permutations, completely determine the adjacencies from 
$z$ to the vertices in 
$X$. But then
\begin{equation*}
(\lvert X \rvert+1)^{t} \geq \lvert\mathcal{N}(X)\rvert.
\end{equation*}Rearranging, we obtain the result.
Unfortunately, while this can yield non-trivial bounds, they are quite a bit weaker than those coming from Theorem 3.1, as 
$\lvert N(X)\rvert \leq n$. On the other hand, 
$\lvert N(X)\rvert$ can be as large as 
$2^{\lvert X\rvert}$. One particular example – where Theorem 3.5 can be seen to give a fairly tight bound is the following: consider a bipartite graph 
$G$ on vertex set 
$(X,Y)$ where 
$\lvert X\rvert = t$, 
$\lvert Y\rvert = 2^{t}$, and the vertices in 
$Y$ are adjacent to the 
$2^{t}$ possible distinct neighbourhoods in 
$X$. Then Theorem 3.5 implies that this graph has 
$\mathcal{R}_{\pi}(G) \geq \frac{t}{\log(t+1)}.$ Theorem 3.8 below implies that 
$\mathcal{R}_{\pi}(G) = O(t),$ so at least for this graph Theorem 3.5 gives a reasonably tight bound.
3.2. Upper bounding representation length
The main purpose in this section is to describe some optimizations to the construction above that prove that the lower bound arising from Theorem 3.1 is nearly tight.
In particular we prove:
Theorem 3.6 Suppose 
$G$ is a graph on 
$n$ vertices. Then
\begin{equation*}
\mathcal{R}_{\pi}(G) \leq 4n + O(\log n).
\end{equation*}The key observation is that one can implement a ‘divide-and-conquer’ approach by realizing two halves of a graph simultaneously, and then ‘shuffle’ one half through the other – essentially in parallel – to realize the edges between the halves. It is this divide and conquer paradigm that allows us to decrease the number of permutations in a realization from quadratic to linear. This is realized through the following lemma.
Lemma 3.7. Suppose 
$G$ is an 
$n$ vertex graph and let 
$X$ and 
$Y$ partition the vertex set of 
$G$. Let 
$G_1$ and 
$G_2$ be the induced graphs on 
$X$ and 
$Y$ respectively. Then
\begin{equation*}
\mathcal{R}_{\pi}(G) \leq \max\{\mathcal{R}_{\pi}(G_1), \mathcal{R}_{\pi}(G_2) \} + 2n.
\end{equation*}Let us derive Theorem 3.6 from Lemma 3.7, then we will return to the proof of the Lemma.
Proof of Theorem 3.6
Let
\begin{equation*} f(n) = \max_{G: \lvert V(G)\rvert=n} \mathcal{R}_{\pi}(G). \end{equation*} Note that 
$f(n)$ is an increasing function of 
$n$ as every 
$n-1$ vertex graph is an induced subgraph of an 
$n$ vertex graph, and a 
$1$-
$11$-representation of a graph 
$G$ contains a 
$1$-
$11$-representation of its 
$n-1$ vertex subgraphs. We prove that
\begin{equation*}
f(n) \leq 4n+2\log_{3/2}(n),
\end{equation*} by induction on 
$n$. This is easy to verify for 
$n \leq 2$, so suppose 
$n \geq 3$. Let 
$G_n$ be a graph on 
$n$ vertices maximizing 
$\mathcal{R}_{\pi}(G)$. Partition 
$V(G_n)$ into parts of size 
$\lfloor n/2 \rfloor$ and 
$\lceil n/2 \rceil$ and let 
$H_1$ and 
$H_2$ be the induced subgraphs on these parts. Then, By Lemma 3.7,
\begin{align*}
f(n)= \mathcal{R}_{\pi}(G_n) &\leq \max\{\mathcal{R}_\pi(H_1), \mathcal{R}_{\pi}(H_2)\} + 2n \\
&\leq \max\{f(\lfloor n/2 \rfloor), f(\lceil n/2 \rceil )\} + 2n \\
& = f(\lceil n/2 \rceil ) + 2n \\
& \leq 4(\lceil n/2 \rceil) + 2\log_{3/2}(\lceil n/2 \rceil) + 2n \\
&\leq 4(n+1)/2 + 2\log_{3/2}(2n/3) + 2n \\
&= 4n + 2\log_{3/2}(n).
\end{align*} Note that here we used the fact that 
$\lceil n/2 \rceil \leq 2n/3$ for 
$n \geq 3$, and this completes the proof.
We now turn to the proof of Lemma 3.7, which follows by modifying the construction of Theorem 2.2.
Proof of Lemma 3.7
Suppose 
$X$ and 
$Y$ partition the vertex set 
$V(G)$, and 
$G_1$ and 
$G_2$ be the graphs induced on 
$X$ and 
$Y$ respectively. Let 
$n_1 = \lvert X\rvert$ and 
$n_2 = \lvert Y\rvert$, so that 
$n_1 + n_2 = n$. Let 
$s = \mathcal{R}_\pi(G_1)$ and 
$t = \mathcal{R}_{\pi}(G_2)$, and assume that without loss of generality that 
$s \geq t$. Then there exists a permutational representation 
$\pi_1 \pi_2 \dots \pi_s$ of 
$G_1$, and a permutational representation 
$\pi_1'\pi_2'\dots\pi_t'$ of 
$G_2$. By Lemma 2.1, this representation of 
$G_2$ can be extended to a representation of length 
$s$ by taking 
$\pi_{t+1}' = \pi_{t+2}' = \dots = \pi_s' = \pi_t'$. One immediately sees then that
\begin{equation*}
w_1 = \pi_1\pi_1'\pi_2\pi_2' \dots \pi_s\pi_s',
\end{equation*} is a word representing 
$G_1$ and 
$G_2$ with a complete bipartite graph between them. We now proceed to augment this word with additional permutations to ‘fix’ the edges between them. The procedure is similar to that illustrated in Theorem 2.2, but designed to take care of adjacencies of multiple vertices simultaneously.
Suppose 
$\pi_{s} = v_1 v_2 \dots v_{n_1}$ and 
$\pi_s' = x_1 x_2 \dots x_{n_2}$. Then for 
$1 \leq i \leq n_1 + n_2 = n$ we define permutations 
$\sigma_i$ and 
$\sigma_{i}'$ as follows:
•
$\sigma_i$ consists of the first
$i$ elements of
$\pi_s'$ being shuffled into the last
$i$ vertices of
$\pi_s$. For
$1 \leq j \leq i$, this means that the element
$x_j$ will be between
$v_{n_1-i+j-1}$ and
$v_{n_1-i+j}$, where
$n_i - i + j \leq 1$ means
$x_j$ precedes all of
$\pi_s$. For any
$x_j$ that precede all of
$\pi_s$, we keep them in their same order as in
$\pi_s'$ – that is
$x_{j-1}$ precedes
$x_j$. For
$j \geq i$, the
$x_j$ come after all elements of
$\pi_s$, again keeping their order.•
$\sigma_i'$ is the same as
$\sigma_i$, except for if
$x_j$ is not adjacent to
$v_{n_1-i+j}$, it is moved to be immediately after
$v_{n_1-i+j}$, while keeping it preceding
$x_{j+1}$.
Thus these two permutations ‘fix’ the edges between the 
$x_{j}$ and 
$v_{n_1-i+j}$ – they add a single instance of a 
$11$ pattern if 
$x_j$ is adjacent to 
$v_{n_1-i+j}$ and two instances if not. We note that when 
$\sigma_{i+1}$ is added, another 
$11$ pattern is added when 
$x_{j}$ is not adjacent to 
$v_{n_1-i+j}$ but not when 
$x_j$ is adjacent to 
$v_{n_1-i+j}$.
Through this procedure we have 
$\sigma_n = \pi_s' \pi_s$ – that is to say, that by 
$\sigma_n'$ all edges have been fixed, and hence
\begin{equation*}
w = \pi_1\pi_1'\pi_2\pi_2' \dots \pi_s\pi_s' \sigma_1\sigma_1'\sigma_2\sigma_2' \dots \sigma_n\sigma_{n}',
\end{equation*} 
$1$-
$11$-represents 
$G$ as desired.
We remark that, if one desires different 
$k$-
$11$-representations, the modifications to Theorem 2.2 can also be adapted to Lemma 3.7 and Theorem 3.6. This will change the constants involved, but the construction will still involve a linear number of permutations (with the constant depending on 
$k$).
The construction in the proof of Lemma 3.7 can also be improved somewhat in the case where one of the graphs is an independent set: in this case after the initial word (
$w_1$ in the proof) is constructed, all edges within the independent set have already been destroyed and we do not need to respect the final permutation of vertices of that graph, when shuffling through the other. A particular application of this idea is the following.
Theorem 3.8 Suppose 
$G$ is a bipartite graph with bipartition 
$(X,Y)$ where 
$\lvert X\rvert \lt \lvert Y\rvert.$ Then
\begin{equation*}
\mathcal{R}_{\pi}(G) \leq 2\lvert X\rvert + 3.
\end{equation*}Proof. Essentially we follow the proof of Lemma 3.7 with minor modifications. Let 
$G_1$ be the independent set on 
$X$ and 
$G_2$ be an independent set on 
$Y$. Let 
$\pi_1$ (respectively 
$\pi_2$) be an arbitrary permutation of vertices in 
$X$ (resp. 
$Y$). Let 
$\pi_1'$ and 
$\pi_2'$ be the reverses of 
$\pi_1$ and 
$\pi_2$, respectively. Then
\begin{equation*}
w_1= \pi_1\pi_2\pi_1'\pi_2'\pi_1\pi_2,
\end{equation*} is easily seen to 
$1$-
$11$-represent the complete bipartite graph on 
$(X,Y)$. Supposing 
$\lvert X\rvert=t$, we now create 
$\sigma_1, \sigma_2, \dots, \sigma_t$ and 
$\sigma'_1, \sigma'_2, \dots, \sigma'_t$ to append the word, to fix the adjacencies between the two partite sets.
If 
$\pi_1 = v_1v_2 \dots v_{t}$, then for 
$1 \leq i \leq t$ we define
•
$\sigma_{i} = v_{1}v_2 \dots v_{t-i} \pi_2 v_{t-i+1} \dots v_t$ to be the permutation obtained by moving all of
$\pi_2$ between
$v_{t-i}$ and
$v_{t-i+1}$.•
$\sigma_i'$ is obtained from
$\sigma_i$ by moving all vertices in
$Y$ not adjacent to
$v_{t-i+1}$ to the right of
$v_{t-i+1}$.
This construction will introduce exactly one 
$11$ pattern between an adjacent pair 
$v_{j} \in X$ and 
$y \in Y$, and at least two instances if not – two when the 
$\sigma_i\sigma'_i$ are added, and one additional when 
$\sigma_{i+1}$ is added. It may add additional 
$11$ patterns between vertices within 
$Y$, but these already are non-edges so no additional edges are deleted. Then the final word is
\begin{equation*}
w_1\sigma_1\sigma_1'\sigma_2\sigma_2'\dots\sigma_t\sigma_t'.
\end{equation*}As noted above, Theorem 3.8 shows the near-sharpness of the example after Theorem 3.5.
4. Conclusion and open problems
In some sense, Theorem 2.2 settles the main question about the class of 
$1$-
$11$-representable graphs. It is, indeed, all graphs. There are still some open questions about 
$1$-
$11$-representations, and related questions, that would be interesting to study.
• Theorem 3.1 gives a lower bound on the number of permutations needed in a permutational representation of some graphs that is close to optimal, but it is ineffective. Is there an easy structural property that implies that
$\mathcal{R}(G)$ or
$\mathcal{R}_{\pi}(G)$ is large? Can one prove, for an explicitly chosen graph, that
$\mathcal{R}_{\pi}(G)$ is large?A natural place to look would be random-like graphs, as random graphs have large
$\mathcal{R}_{\pi}(G)$ with high probability. Perhaps one can show that the Paley graphs have large
$\mathcal{R}_{\pi}(G)$ as this family is known to be quasi-random in the sense of Chung, Graham, and Wilson [Reference Chung, Graham and Wilson3].• While Theorem 3.6 cannot be significantly improved – it is tight within a factor of
$O(\log n)$ – perhaps it can be improved in some cases.For many families of graphs (e.g. bipartite graphs, graphs with chromatic number exactly
$k$, split graphs,
$\dots$) the number of graphs in the family of order
$n$ grows like
$2^{c n^{2}}$ and in these cases, our counting bound adapts to give lower bounds a similar
$O(\log n)$ factor away from our upper bound.The family of planar graphs seems particularly interesting. There are only exponentially many planar graphs and counting essentially gives no non-trivial lower bound. Can one either find planar graphs with high
$R_{\pi}(G)$ or prove an
$o(n)$ or even
$O(n^{1-\epsilon})$ upper bound for
$\mathcal{R}_{\pi}(G)$ for graphs in this class?• The example of
$H_{n,n}$ shows that sometimes the word-representation number can be significantly larger than
$R_\pi(G)$ of the same graph. This is despite the fact that, in a sense, one needs more effort to encode non-edges. It would be interesting to study when this happens: what causes a graph to be easier to
$k$-
$11$-represent than it is to
$(k-1)$-
$11$-represent?
Acknowledgements
We would like to thank Andrew Owens and Travis Pence for helpful discussions in early stages of this work. Portions of this research were conducted at the Masamu Advanced Studies Institute (MASI) in Windhooek, Namibia and we gratefully acknowledge NSF grant NSF-DMS 2015425. We would also like to thank the anonymous referees, whose helpful comments have helped clarify the paper and simplify the proof of Lemma 2.5.
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