# Bounds for boxicity of some classes of graphs

T. Kavaskar  Department of Mathematics, Central University of Tamil Nadu,
Thiruvarur-610005, India<EMAIL_ADDRESS>
###### Abstract.

Let $box(G)$ be the boxicity of a graph $G$, $G[H_{1},H_{2},\ldots,H_{n}]$ be
the $G$-generalized join graph of $n$-pairwise disjoint graphs
$H_{1},H_{2},\ldots,H_{n}$, $G^{d}_{k}$ be a circular clique graph (where
$k\geq 2d$) and $\Gamma(R)$ be the zero-divisor graph of a commutative ring
$R$. In this paper, we prove that $\chi(G^{d}_{k})\geq box(G^{d}_{k})$, for
all $k$ and $d$ with $k\geq 2d$. This generalizes the results proved in [3].
Also we obtain that
$box(G[H_{1},H_{2},\ldots,H_{n}])\leq\mathop{\sum}\limits_{i=1}^{n}box(H_{i})$.
As a consequence of this result, we obtain a bound for boxicity of zero-
divisor graph of a finite commutative ring with unity. In particular, if $R$
is a finite commutative non-zero reduced ring with unity, then
$\chi(\Gamma(R))\leq box(\Gamma(R))\leq 2^{\chi(\Gamma(R))}-2$. where
$\chi(\Gamma(R))$ is the chromatic number of $\Gamma(R)$. Moreover, we show
that if
$N=\prod\limits_{i=1}^{a}p_{i}^{2n_{i}}\prod\limits_{j=1}^{b}q_{j}^{2m_{j}+1}$
is a composite number, where $p_{i}$’s and $q_{j}$’s are distinct prime
numbers, then
$box(\Gamma(\mathbb{Z}_{N}))\leq\big{(}\mathop{\prod}\limits_{i=1}^{a}(2n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(2m_{j}+2)\big{)}-\big{(}\mathop{\prod}\limits_{i=1}^{a}(n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(m_{j}+1)\big{)}-1$,
where $\mathbb{Z}_{N}$ is the ring of integers modulo $N$. Further, we prove
that, $box(\Gamma(\mathbb{Z}_{N}))=1$ if and only if either $N=p^{n}$ for some
prime number $p$ and some positive integer $n\geq 2$ or $N=2p$ for some odd
prime number $p$.

###### Key words and phrases:

Boxicity, Circular clique graph, Generalized join of graphs, Zero-Divisor
Graph.

###### 2010 Mathematics Subject Classification:

05C62, 05C15.

##  1\. Introduction

In this paper, all graphs considered are simple and undirected. Given a non-
empty set $\Upsilon$ and a collection $\mathbf{D}$ of subsets of $\Upsilon$,
the intersection graph of $\mathbf{D}$ is defined as the graph with vertex set
$\mathbf{D}$ such that two elements of $\mathbf{D}$ are adjacent if and only
if their intersection is non-empty. An $\ell$-box is the Cartesian product
$[x_{1},\ y_{1}]\times\ldots\times[x_{\ell},\ y_{\ell}]$ of $\ell$ closed and
bounded intervals of the real line. The boxicity of a graph
$G=\left(V(G),E(G)\right)$, denoted by $box(G)$, is the least positive integer
$\ell$ such that $G$ is isomorphic to the intersection graph of a family of
$\ell$-boxes in Euclidean $\ell$-space. From a computational point of view,
the boxicity of a graph is an NP-hard parameter, which was proved by
Yannakakis [29]. However, various bounds have been found in terms of the
maximum degree, acylic chromatic number, weak $r$-coloring number, genus, etc.
See for instance [1], [11], [12], [14]-[18]. Boxiity is used as a measure of
the complexity of ecological [25] and social [19] networks, and has
applications in fleet maintenance [23]. In [12], S. Chandran et al., proved
that $box(G)\leq 2\Delta^{2}(G)$ for any graph $G$, where $\Delta(G)$ is the
maximum degree of $G$. This bound was improved by L. Esperet in [17] to
$box(G)\leq\frac{\left\lfloor\Delta(G)^{2}\right\rfloor}{2}+2$.

For a given finite class of graphs $H_{i}=(V,E_{i})$, $i=1,2,\ldots,k$ with
same vertex set $V$, the intersection $H_{1}\cap H_{2}\cap\ldots\cap H_{k}$ of
the graphs $H_{1},H_{2},\ldots,H_{k}$ is the graph with vertex set $V$ and
edge set $E_{1}\cap E_{2}\cap\ldots\cap E_{k}$. A graph $G$ is said to be an
interval graph if there is a function $f$ from the vertex set of $G$ into a
set of closed and bounded intervals in the real line such that $uv\in E(G)$ if
and only if $f(u)\cap f(v)\neq\emptyset$. Such a function $f$ is called an
interval representation of $G$. Note that $box(G)=1$ if and only if $G$ is an
interval graph.

Let us recall that a result proved by Roberts in [24].

######  Theorem 1.1 ([24]).

Let $G$ be a graph. Then $box(G)\leq\ell$ if and only if there exist $\ell$
interval graphs $I_{1},\ldots,I_{\ell}$ with vertex set $V(G)$ such that
$E(G)=E(I_{1})\cap\ldots\cap E(I_{\ell})$ holds.

For a given $S\subseteq V(G)$, the subgraph of $G$ induced by $S$ is denoted
by $\langle S\rangle$. A clique in $G$ is a complete subgraph of $G$. The
clique number of a graph $G$, denoted by $\omega(G)$, is the largest size of a
clique in $G$. A subset $U\subseteq V(G)$ is said to be independent if no two
vertices in $U$ are adjacent in $G$.

Let $k$ be a positive integer. A proper $k$-coloring of a graph $G$ is a
function from $V(G)$ into the set of $k$ colors such that no two adjacent
vertices receive the same color. The chromatic number of a graph $G$, denoted
by $\chi(G)$, is the least positive integer $k$ such that there exists a
proper $k$-coloring of $G$.

Let $k$ and $d$ be two positive integers with $k\geq 2d$. The well-known
circular clique $G^{d}_{k}$ has the vertex set $V(G^{d}_{k})=\\{a_{j}\ |\
0\leq j\leq k-1\\}$ and the edge set $E(G^{d}_{k})=\\{a_{i}a_{j}|\
d\leq|i-j|\leq k-d\\}$. The circular clique graph $G^{d}_{k}$ is a special
class of circulant graphs which play an important role in the study of
circular chromatic number. This can be seen in [31, 32]. Note that, the
circular clique graph is vertex-transitive.

In [9], Bhowmick et al., proved that $box(G)\leq\chi(G)$ holds for any graph
$G$ with no asteroidal triples. This result is not true in general. For
example, the graph $G$ obtained from a complete bipartite graph with $2n$
(where $n\geq 5$) vertices by removing a perfect matching has
$box(G)=\left\lceil\frac{n}{4}\right\rceil>2=\chi(H)$ (see [10]). S. Charndran
et al., in [13] observed that almost all graphs have boxicity more than their
chromatic number by using probabilistic method. Interestingly, the family of
graphs with boxicity at most their chromatic number is also not narrow. So it
is interesting to characterize graphs $G$ for which $\chi(G)\geq box(G)$
holds. In this direction, Akira Kamibeppu [3] gave a sufficient condition for
$box(G)\leq\chi(G)$. Moreover they showed that $box(G)\leq\chi(G)$ hold for a
special class of circular clique $G^{d}_{k}$ with an asteroidal triple.

######  Theorem 1.2 ([3]).

For a circulant graph $G^{d}_{nd}$ with $n\geq 2$ and $d\geq 1$, the
inequality $\chi(G^{d}_{nd})\geq box(G^{d}_{nd})$ holds.

######  Theorem 1.3 ([3]).

For a circulant graph $G^{d}_{nd+r}$ with $n\geq 2$, $d\geq 2$ and $1\leq
r<d$, if $n\geq d-r-1$, the inequality $\chi(G^{d}_{nd+r})\geq
box(G^{d}_{nd+r})$ holds.

We recall the definition of $G$-generalized join of graphs ([27]). Let $G$ be
a graph with $V(G)=\\{u_{1},u_{2},\ldots,u_{n}\\}$ and
$H_{1},H_{2},\ldots,H_{n}$ be pairwise disjoint graphs. The $G$-generalized
join $G[H_{1},H_{2},\ldots,H_{n}]$ of $H_{1},H_{2},\ldots,H_{n}$ is the graph
obtained by replacing each vertex $u_{i}$ of $G$ by $H_{i}$ and joining each
vertex of $H_{i}$ to each vertex of $H_{j}$ if $u_{i}$ is adjacent to $u_{j}$
in $G$. Note that, if $G=K_{2}$, then the $G$-generalized join
$K_{2}[H_{1},H_{2}]$ of $H_{1}$ and $H_{2}$ is the join $H_{1}\vee H_{2}$ of
$H_{1}$ and $H_{2}$.

In [6], an equivalence relation $\mathtt{\sim}_{G}$ for any graph $G$ was
introduced as follows: For any $x,y\in V(G)$, $x\mathtt{\sim}_{G}y$ if and
only if $N(x)=N(y)$. Let $[x]$ denote the equivalence class which contains
$x$, and $S=\\{[x_{1}],[x_{2}],\ldots,[x_{k}]\\}$ be the set of all
equivalence classes of $\mathtt{\sim}_{G}$. Clearly, the set of vertices in a
class $[x]$ is an independent subset of $G$. The reduced graph of $G$ is a
simple graph $G_{r}$ with vertex set $V(G_{r})=S$ and two distinct vertices
$[x]$ and $[y]$ are adjacent in $G_{r}$ if and only if $x$ and $y$ are
adjacent in $G$. Note that, any graph $G$ is the $G_{r}$-generalized join of
$\left\langle[x_{1}]\right\rangle,\ldots,\left\langle[x_{k}]\right\rangle$,
that is,
$G=G_{r}\big{[}\left\langle[x_{1}]\right\rangle,\ldots,\left\langle[x_{k}]\right\rangle\big{]}$.

Throughout this paper, $R$ denotes a non-zero finite commutative ring with
unity 1. A non-zero element $x$ of $R$ is said to be a zero-divisor if there
exists a non-zero element $y$ of $R$ such that $xy$=0. An element $u$ of $R$
is unit in $R$ if there exists $v$ in $R$ such that $uv$=1. The nilradical of
a ring $R$ is the set $J=\\{x\in R:x^{t}=0,$ for some positive integer $t\\}$.
A ring $R$ is said to be reduced if $J=\\{0\\}$. For any ring $R$, in [8],
Beck associated a simple graph with $R$ whose vertices are the elements of $R$
and any two distinct vertices $x$ and $y$ are adjacent if and only if $xy$=0
in $R$. It can be observed that for the graph associated with the ring, the
vertex $0$ is adjacent with every other vertex. D.F. Anderson et. al. in [7]
slightly modified the definition of the graph associated with a ring by
considering the nonzero zero divisors as the vertices and any two distinct
vertices $x$ and $y$ are adjacent if and only if $xy=0$ in $R$. They called
this as the zero-divisor graph of the ring and it is denoted by $\Gamma(R)$.
The zero-divisor graph have been studied extensively by several authors. See
for instance [2]-[7], [16], [20], [21].

For $x,y\in R$, define $x\mathtt{\sim}_{R}y$ if and only if $Ann(x)=Ann(y)$.
It is proved in [22] that the relation $\mathtt{\sim}_{R}$ is an equivalence
relation on $R$. For $x\in R$, let $C_{x}=\\{r\in R\ |\
x\mathtt{\sim}_{R}r\\}$ be the equivalence class of $x$. Let
$R_{E}=\\{C_{x_{1}},C_{x_{2}},\ldots,C_{x_{k}}\\}$ be the set of equivalence
classes of the relation $\mathtt{\sim}_{R}$ other than $C_{0}$ and $C_{1}$.
The compressed zero-divisor graph $\Gamma_{E}(R)$ (defined in [28]) is the
simple graph with vertex set $R_{E}$and two distinct vertices $C_{x}$ and
$C_{y}$ are adjacent if and only if $xy$=0. The following result can be found
in [26].

######  Theorem 1.4 ([26]).

If $R$ is a finite commutative ring with unity, then
(i) $\Gamma(R)\cong\Gamma_{E}(R)[\left\langle
C_{x_{1}}\right\rangle,\left\langle C_{x_{2}}\right\rangle,\ldots,\left\langle
C_{x_{k}}\right\rangle]$,
(ii) $\left\langle C_{x_{i}}\right\rangle$ is complete if and only if
$x_{i}^{2}=0$, and
(iii) $\left\langle C_{x_{i}}\right\rangle$ is totally disconnected (that is,
complement of complete graph) if and only if $x_{i}^{2}\neq 0$.

This paper is organized as follows. In the second section, we prove that
$\chi(G^{d}_{k})\geq box(G^{d}_{k})$, for any $k\geq 2d$ and as a consequence
we have Theorems 1.2 and 1.3. In fact, this was a question raised in [3].

In the third section, we obtain $M\leq
box\left(G[H_{1},H_{2},\ldots,H_{n}]\right)\leq\mathop{\sum}\limits_{i=1}^{n}box(H_{i}),$
if at least one of $H_{i}$’s is not complete and
$M=\max\Big{\\{}\mathop{\sum}\limits_{j=1}^{k}box(H_{i_{j}})\
|\\{u_{i_{1}},\ldots,u_{i_{k}}\\}$ is a clique in $G$ and each $H_{i_{j}}$ is
not a complete graph$\Big{\\}}$. As a consequence of this result, we prove
that if $G_{r}$ is the reduced graph of $G$, then $box(G)\leq|V(G_{r})|$.

In the fourth section, we obtain a bound for the boxicity of zero-divisor
graph of a ring. In particular, we prove that $\chi(\Gamma(R))\leq
box(\Gamma(R))\leq 2^{\chi(\Gamma(R))}-2$ if $R$ is a non-zero reduced ring.

In the last section, for a composite number $N$, we prove that if
$N=\prod\limits_{i=1}^{a}p_{i}^{2n_{i}}\prod\limits_{j=1}^{b}q_{j}^{2m_{j}+1}$,
where $p_{i}$’s and $q_{j}$’s are distinct prime numbers, then
$\omega(\Gamma_{E}(\mathbb{Z}_{N}))=\mathop{\prod}\limits_{i=1}^{a}(n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(m_{j}+1)+b-1=\chi(\Gamma_{E}(\mathbb{Z}_{N}))$,
where $\mathbb{Z}_{n}$ is a ring of integer modulo $N$ and

$box(\Gamma(\mathbb{Z}_{N}))\leq\left(\mathop{\prod}\limits_{i=1}^{a}(2n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(2m_{j}+2)\right)-\left(\mathop{\prod}\limits_{i=1}^{a}(n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(m_{j}+1)\right)-1.$

Finally, we have shown that $box(\Gamma(\mathbb{Z}_{N}))=1$ if and only if
either $N=p^{n}$ for some prime number $p$ and some positive integer $n\geq 2$
or $N=2p$ for some odd prime number $p$.

##  2\. Boxicity of Circular Clique $G^{d}_{k}$

We recall the following result from [31, 32].

######  Theorem 2.1 ([31, 32]).

If $k$ and $d$ are any two positive integers with $k\geq 2d$, then
$\chi(G^{d}_{k})=\left\lceil\frac{k}{d}\right\rceil$.

Using Theorems 1.1 and 2.1, we prove Theorem 2.2.

######  Theorem 2.2.

If $k$ and $d$ are any two positive integers with $k\geq 2d$, then
$\chi(G^{d}_{k})\geq box(G^{d}_{k})$.

###### Proof.

Let $k=md+b$, where $0\leq b\leq d-1$.

Case 1. $m\geq 3$.

Let $r=d$ or $b$ and let $W=\\{a_{j}\ |\ 0\leq j\leq r-1\\}$. Then $W$ is an
independent subset of $G^{d}_{k}$. We now claim the following for $W$.
Claim 1. There is an interval graph $I_{W}$ such that $G^{d}_{k}$ is a
spanning subgraph of $I_{W}$ and $W$ is an independent subset of $I_{W}$.
Proof of claim 1.

Let $H$ be the complete graph with the vertex set

$V(H)=\\{a_{j}\ |\ r\leq j\leq d-1\\}\cup\\{a_{d+j}\ |\ r\leq j\leq k-1-d\\}$

and $S$ be the complete graph with the vertex set $V(S)=\\{a_{d+j}\ |\ 0\leq
j\leq r-1\\}$. Define $I_{W}$ to be the graph with the vertex set
$V(I_{W})=V(G^{d}_{k})$ and the edge set

$E(I_{W})=E(G^{d}_{k})\cup E(H)\cup E(S)\cup\\{a_{i}a_{d+j}\ |\ 0\leq i\leq
r-1,r\leq j\leq k-d-1\\}$

$\cup\\{a_{i}a_{d+j}|\ r\leq i\leq d-1$, or $d+r\leq i\leq k-1\text{ and
}0\leq j\leq r-1\\}.$
Then clearly $G^{d}_{k}$ is a spanning subgraph of $I_{W}$, $W$ is an
independent subset of $I_{W}$ and there is no edge between $W$ and
$\\{a_{j}:r\leq j\leq d-1\\}$ in $I_{W}$.

We shall now show that $I_{W}$ is our required interval graph. The following
is an interval representation $f$ of $I_{W}$.
(i) For $0\leq i\leq r-1$,

$f(a_{i})=\\{d-i\\},$

and

$f(a_{d+i})=[d-i,\ d+1],$

(ii) For $r\leq i\leq d-1$,

$f(a_{i})=\\{d+1\\},$

(iii) For $d+r\leq i\leq k-1$,

$f(a_{i})=[1,\ d+1].$

Note that for $1\leq i\leq r-1$, the number $d-i$ does not belong to the
interval $[d-j,d+1]$, for $\ 0\leq j\leq i-1$.

Let $x,y\in V(I_{W})$, say $x=a_{i}$ and $y=a_{j}$ where $i<j$. First, if
$xy\notin E(I_{W})$, then either $x,y\in W$, or $x\in W$ and
$y\in\\{a_{r},a_{r+1},\ldots,a_{d-1}\\}$ or $x\in W$ and $y\in S$, where
$j\in\\{d+1,d+2,\ldots,d+i-1\\}$ (as $m\geq 3$). We can observe in all the
case that $f(x)\cap f(y)=\emptyset$. Next, if $xy\in E(I_{W})$, then either
$x,y\in S\cup V(H)$ or $x\in W$ and $y\in S\cup V(H)$. In both the cases
$f(x)\cap f(y)\neq\emptyset$. Thus $I_{W}$ is an interval graph with the
desired property.

Subcase (i) $b\geq 1$.

By Theorem 2.1, $\chi(G^{d}_{k})=m+1$. Let
$V(G^{d}_{k})=\mathop{\bigcup}\limits_{i=0}^{m}W_{i}$, where
$W_{i}=\\{a_{id+j}\ |\ 0\leq j\leq d-1\\}$, for $0\leq i\leq m-1$ and
$W_{m}=\\{a_{md+j}\ |\ 0\leq j\leq b-1\\}$. Clearly, $W_{i}\cap
W_{j}=\emptyset$, for $0\leq i,j\leq m$ with $i\neq j$ and $W_{i}$ is an
independent subset of $G^{d}_{k}$, for $0\leq i\leq m$. Since $G_{k}^{d}$ is a
vertex-transitive graph, for each $0\leq i\leq m$, there is an interval graph
$I_{W_{i}}$ for $W_{i}$ such that $G^{d}_{k}$ is a spanning subgraph of
$I_{W_{i}}$ and $W_{i}$ is an independent subset of $I_{W_{i}}$, by Claim 1.

By Theorem 1.1, it is enough to prove
$E(G^{d}_{k})=\mathop{\bigcap}\limits_{i=0}^{m}E(I_{W_{i}})$. Let $x,y\in
V(G^{d}_{k})$. If they are adjacent in $G^{d}_{k}$, then they are adjacent in
each $I_{W_{i}}$, for $0\leq i\leq m$, because $G^{d}_{k}$ is a spanning
subgraph of $I_{W_{i}}$. It remains to show that if $x$ and $y$ are not
adjacent in $G^{d}_{k}$, then they are not adjacent in at least one of
$I_{W_{i}}$’$s$. Suppose if $x,y\in W_{i}$, for some $i$, then they are not
adjacent in $I_{W_{i}}$. Therefore, by the definition of $G^{d}_{k}$, $x\in
W_{i}$ and $y\in W_{i+1}$, for some $i$ (where the addition in subscript is
taken modulo over $m$), then they are not adjacent in $I_{W_{i}}$ or in
$I_{W_{i+1}}$. Hence $\chi(G^{d}_{k})\geq box(G^{d}_{k})$.

Subcase (ii) $b=0$.

By Theorem 2.1, $\chi(G^{d}_{k})=m$. Let
$V(G^{d}_{k})=\mathop{\bigcup}\limits_{i=0}^{m-1}W_{i}$, where
$W_{i}=\\{a_{id+j}\ |\ 0\leq j\leq d-1\\}$, for $0\leq i\leq m-1$. By a
similar strategy as in Subcase (i), we can prove that $\chi(G^{d}_{k})\geq
box(G^{d}_{k})$.

Case 2. $m=2$.

Then $k=2d+b$, where $0\leq b\leq d-1$. Suppose if $b=0$, then $G^{d}_{2d}$ is
an $1$-regular graph (that is, the degree of every vertex of $G^{d}_{2d}$ is
one) and hence it is an interval graph. Thus,
$box(G^{d}_{2d})=1<2=\chi(G^{d}_{2d})$. So, let us assume that $b\geq 1$.

First if $U=\\{a_{j}\ |\ 0\leq j\leq b-1\\}$, then by the similar way as given
in Claim 1, we can construct an interval graph $I_{U}$ for $U$ such that
$G^{d}_{k}$ is a spanning subgraph of $I_{U}$ and $U$ is an independent subset
of $I_{U}$. Next if $W=\\{a_{j}\ |\ 0\leq j\leq d-1\\}$, then we claim the
following for $W$.

Claim 2. There is an interval graph $I_{W}$ such that $G^{d}_{k}$ is a
spanning subgraph of $I_{W}$ and $W$ is an independent subset of $I_{W}$.
Proof of claim 2.
Since $2\leq b+1\leq d$, there exist non-negative integers $c,e$ such that
$d=c(b+1)+e$ where $0\leq e\leq b$. Let $S=\\{a_{d+j}\ |\ 0\leq j\leq d-1\\}$.
We write $S=\mathop{\bigcup}\limits_{i=0}^{c}S_{i}$, where
$S_{i}=\\{a_{d+i(b+1)+j}\ |\ 0\leq j\leq b\\}$ for $0\leq i\leq c-1$ and
$S_{c}=\\{a_{d+c(b+1)+j}\ |\ 0\leq j\leq e-1\\}$.

Let $H$ be the complete graph with vertex set $V(H)=\\{a_{2d+j}\ |\ 0\leq
j\leq b-1\\}$ and $S^{\prime}$ be a graph with vertex set $V(S^{\prime})=S$
and edge set $E(S^{\prime})=\mathop{\bigcup}\limits_{i=0}^{c}E_{i}$, where

$\displaystyle E_{i}$ $\displaystyle=$ $\displaystyle\\{a_{d+i(b+1)+j}\
a_{d+i(b+1)+\ell}\ |\ 0\leq j<\ell\leq b\\}$
$\displaystyle\cup\\{a_{d+(i+1)(b+1)+j}\ a_{d+i(b+1)+\ell}\ |\ 0\leq j\leq b,\
j\leq\ell\leq b\\},$

for $0\leq i\leq c-2$,

$\displaystyle E_{c-1}$ $\displaystyle=$ $\displaystyle\\{a_{d+(c-1)(b+1)+j}\
a_{d+(c-1)(b+1)+\ell}\ |\ 0\leq j<\ell\leq b\\}$
$\displaystyle\cup\\{a_{d+c(b+1)+j}\ a_{d+(c-1)(b+1)+\ell}\ |\ 0\leq j\leq
e-1,\ j\leq\ell\leq b\\}.$

and $E_{c}=\\{a_{d+c(b+1)+j}\ a_{d+c(b+1)+\ell}\ |\ 0\leq j<\ell\leq e-1\\}.$

Define $I_{W}$ be the graph with $V(I_{W})=V(G^{d}_{k})$ and

$E(I_{W})=E(G^{d}_{k})\cup E(H)\cup E(S^{\prime})\cup\\{a_{i}a_{j}|\ 0\leq
i\leq 2d-1,2d\leq j\leq k-1\\},$

that is, $E(I_{W})=E(G^{d}_{k})\cup E(\langle W\vee H\rangle)\cup E(\langle
S^{\prime}\vee H\rangle).$ Then clearly $G^{d}_{k}$ is a spanning subgraph of
$I_{W}$ and $W$ is an independent subset of $I_{W}$.

Now we prove that $I_{W}$ is an interval graph. The following is an interval
representation $f$ of $I_{W}$.

For $0\leq i\leq d-1$ (that is, $a_{i}\in W$),

(1) $\displaystyle f(a_{i})=\\{\alpha_{i}\\},\text{where
}\alpha_{i}\in(i-1,i),$

For $0\leq j\leq b$ (that is, $a_{d+j}\in S_{0}$),

(2) $\displaystyle f(a_{d+j})=[-1,\ j],$

for $1\leq i\leq c-1$ and $0\leq j\leq b$ (that is, $a_{d+i(b+1)+j}\in S_{i}$)

(3) $\displaystyle f(a_{d+i(b+1)+j})=[(i-1)(b+1)+j,\ i(b+1)+j],$

and for $0\leq j\leq e-1$ (that is, $a_{d+c(b+1)+j}\in S_{c}$),

(4) $\displaystyle f(a_{d+c(b+1)+j})=[(c-1)(b+1)+j,\ c(b+1)+j],$

For $2d\leq i\leq k-1$, (that is, $a_{i}\in V(H)$)

(5) $\displaystyle f(a_{i})=[-1,\ d].$

Clearly, $f(x)\subseteq[-1,d]$, for every $x\in V(I_{W})$.

Let $x,y\in V(I_{W})$. First if $xy\notin E(I_{W})$, then $x,y\in W\cup S$.
More precisely, one of the following will happen. Let us establish that
$f(x)\cap f(y)=\emptyset$ in all the possibilities.

  * •

$x,y\in W$. Then there exist $0\leq i<j\leq d-1$ such that $x=a_{i}$ and
$y=a_{j}$. Hence $f(x)\cap
f(y)=\\{\alpha_{i}\\}\cap\\{\alpha_{j}\\}=\emptyset$ (by equations (2)).

  * •

$x\in W$ and $y\in S$ or $y\in W$ and $x\in S$. Without loss of generality,
let $x\in W$ and $y\in S$. Then $x=a_{i}$ for some $i\in\\{0,1,\ldots,d-1\\}$
and $y=a_{d+j}$, for some $j\in\\{0,1,\ldots,i-1\\}$ or
$j\in\\{b+i+1,\ldots,d-1\\}$. Clearly, $f(x)=\\{\alpha_{i}\\}\subseteq(i-1,i)$
(by equation (2)). When $0\leq j\leq i-1$,
$f(y)=f(a_{d+j})\subseteq[-1,j]\subseteq[-1,i-1]$ (by equations (2), (3) and
(4)) and when $b+i+1\leq j\leq d-1$,
$f(y)=f(a_{d+j})=[j-b-1,j]\subseteq[i,d-1]$ (by equations (3) and (4)). In
both the cases $f(x)\cap f(y)=\emptyset$.

  * •

$x,y\in S$. Then $x\in S_{i}$ and $y\in S_{j}$ for some $0\leq i,j\leq c$.
Since $S_{i}$’s are complete, $i\neq j$, say $i<j$.

First, let us consider $j\geq i+2$. Clearly, $f(x)\subseteq[-1,i(b+1)+b]$ and
$f(y)\subseteq[(j-1)(b+1),j(b+1)+b]$ and hence $f(x)\cap f(y)=\emptyset$.

Next, let us consider $j=i+1$. When $0\leq i\leq c-2$, $x=a_{d+i(b+1)+\ell}$
for some $0\leq\ell<b$ and $y=a_{d+(i+1)(b+1)+\ell+p}$ for some $1\leq p\leq
b-\ell$. One can see that $f(x)\subseteq[-1,\ i(b+1)+\ell]$ and
$f(y)=[i(b+1)+\ell+p,\ (i+1)(b+1)+\ell+p]$ and thus $f(x)\cap f(y)=\emptyset$,
as $p\geq 1$. Similarly when $i=c-1$, we can show that $f(x)\cap
f(y)=\emptyset$.

Next if $xy\in E(I_{W})$, then one of the following will happen.

  * •

$x\in V(H)$ and $y\in V(I_{W})$ or vice versa.

  * •

$x,y\in S_{i}$, for some $i\in\\{0,1,\ldots,c\\}$

  * •

$x\in S_{i}$ and $y\in S_{i+1}$, for some $i\in\\{0,1,\ldots,c\\}$ or vice
versa.

  * •

$x\in W$ and $y\in S$ or vice versa.

By using the definition of $f$, without much difficulty, one can see that
$f(x)\cap f(y)\neq\emptyset$. Thus, $I_{W}$ is an interval graph. Hence the
Claim 2.
By Theorem 2.1, $\chi(G^{d}_{k})=3$. Let
$V(G^{d}_{k})=\mathop{\bigcup}\limits_{i=0}^{2}W_{i}$, where
$W_{i}=\\{a_{id+j}\ |\ 0\leq j\leq d-1\\}$, for $0\leq i\leq 1$ and
$W_{2}=\\{a_{2d+j}\ |\ 0\leq j\leq b-1\\}$. Here also similar to Case 1, we
can prove that $\chi(G^{d}_{k})=3\geq box(G^{d}_{k})$. ∎

##  3\. Bounds for Boxicity of $G$-Generalized Join of Graphs

In this section we first obtain a bounds for boxicity of $G$-generalized join
of graphs.

######  Theorem 3.1.

Let $G$ be a graph with vertex set $V(G)=\\{u_{1},\ldots,u_{n}\\}$ and
$H_{1},\ldots,H_{n}$ be $n$-pairwise disjoint graphs such that at least one of
$H_{i}$’s is not complete. Then $M\leq
box(G[H_{1},H_{2},\ldots,H_{n}])\leq\mathop{\sum}\limits_{i=1}^{n}box(H_{i})$,
where $M=\max\Big{\\{}\mathop{\sum}\limits_{j=1}^{k}box(H_{i_{j}})\
|\\{u_{i_{1}},\ldots,u_{i_{k}}\\}$ is a clique in $G$ and each $H_{i_{j}}$ is
not a complete graph$\Big{\\}}$.

###### Proof.

It is easy to note that

  1. (i)

if $G$ and $H$ are not complete graphs, then $box(G\vee H)\geq box(G)+box(H)$;

  2. (ii)

if $G^{\prime}$ is an induced subgraph of $G$, then $box(G)\geq
box(G^{\prime})$.

If $\langle\\{u_{i_{1}},\ldots,u_{i_{k}}\\}\rangle$ is a clique in $G$,
$H_{i_{j}}$ is not a complete graph, for $1\leq j\leq k$, then
$G[H_{i_{1}},H_{i_{2}},\ldots,H_{i_{k}}]=H_{i_{1}}\vee H_{i_{2}}\vee\ldots\vee
H_{i_{k}}$ is an induced subgraph of $G[H_{1},H_{2},\ldots,H_{n}]$. Thus by
(i) and (ii),

$box(G[H_{1},H_{2},\ldots,H_{n}])\geq
box(G[H_{i_{1}},H_{i_{2}},\ldots,H_{i_{k}}])\geq\mathop{\sum}\limits_{j=1}^{k}box(H_{i_{j}}),$

by induction on $k$ and hence the lower bound follows.

Next, we prove the upper bound. For $1\leq i\leq n$, let $box(H_{i})=k_{i}$.
Then, by Theorem 1.1, there exist interval graphs
$I_{i}^{1},\ldots,I_{i}^{k_{i}}$ with vertex set $V(H_{i})$ such that
$H_{i}=\mathop{\bigcap}\limits_{j=1}^{k_{i}}I_{i}^{j}$. We now define an
interval graph $J_{i}^{j}$ such that $G[H_{1},H_{2},\ldots,H_{n}]$ is a
spanning subgraph of $J_{i}^{j}$, for all $1\leq i\leq n$ and $1\leq j\leq
k_{i}$.

For $1\leq i\leq n$, denote by $N_{G}(u_{i})$ the set of all vertices
$u_{i_{1}},u_{i_{2}},\ldots u_{i_{t}}$ of $G$ which are adjacent to $u_{i}$ in
$G$. Let $R_{i}$ be the graph with vertex set $V(G)$ and edge set

$E(R_{i})=\\{xu_{i}\ |\ x\in N_{G}(u_{i})\\}\cup\\{xy\ |\ x\neq u_{i}\mbox{
and }y\neq u_{i}\\}.$

Then $G$ is a spanning subgraph of every $R_{i}$. For every $1\leq\ell\leq n$
with $\ell\neq i$, let $S_{\ell}$ be the complete graph with vertex set
$V(H_{\ell})$. Define
$J_{i}^{j}=R_{i}[S_{1},\ldots,S_{i-1},I_{i}^{j},S_{i+1},\ldots,S_{n}]$, for
$1\leq i\leq n$ and $1\leq j\leq k_{i}$. Clearly $G[H_{1},H_{2},\ldots,H_{n}]$
is a spanning subgraph of $J_{i}^{j}$.

Claim 1. $J_{i}^{j}$ is an interval graph, for $1\leq i\leq n$ and $1\leq
j\leq k_{i}$.

Proof of Claim 1. Let $c_{i}^{j}$ be an interval representation of
$I_{i}^{j}$. Then there exist real numbers $x$ and $y$ such that
$\mathop{\bigcup}\limits_{v\in V(I_{i}^{j})}c_{i}^{j}(v)$ is contained in
$[x,y)$. We construct an interval representation $f_{i}^{j}$ of $J_{i}^{j}$ as
follows.

$f_{i}^{j}(v)=\begin{cases}[x,y]&\mbox{if $v\in S_{i_{\gamma}}$,
$1\leq\gamma\leq t$}\\\ [y,z]&\mbox{if $v\in S_{\mu}$,
}\mbox{$\mu\in\\{1,2,\ldots,n\\}\setminus\\{i,i_{1},\ldots,i_{t}\\}$,}\\\
c_{i}^{j}(v)&\mbox{if $v\in I_{i}^{j}$.}\end{cases}$

First, note that if $vw\notin E(J_{i}^{j})$, then either $v,w\in V(I_{i}^{j})$
or $v\in V(I_{i}^{j})$ and $w\in S_{\mu}$, for some
$\mu\in\\{1,2,\ldots,n\\}\setminus\\{i,i_{1},\ldots,i_{t}\\}$, say. In the
first case, $f_{i}^{j}(v)\cap f_{i}^{j}(w)=c_{i}^{j}(v)\cap
c_{i}^{j}(w)=\emptyset$, and the later case, the interval
$f_{i}^{j}(v)=c_{i}^{j}(v)$ is a subset of $[x,y)$ and $f_{i}^{j}(w)=[y,z]$
and hence $f_{i}^{j}(v)\cap f_{i}^{j}(w)=\emptyset$.

If $vw\in E(J_{i}^{j})$, then either $v,w\in V(I_{i}^{j})$ or $v\in
V(I_{i}^{j})$ and
$w\in\mathop{\bigcup}\limits_{\gamma=1}^{t}V(S_{i_{\gamma}})$, or $v,w\not\in
V(I_{i}^{j})$. In all cases, $f_{i}^{j}(v)\cap f_{i}^{j}(w)\neq\emptyset$.
Hence the claim 1.

Claim 2.
$E(G[H_{1},\ldots,H_{n}])=\mathop{\bigcap}\limits_{i=1}^{n}\mathop{\bigcap}\limits_{j=1}^{k_{i}}E(J_{i}^{j})$.

As $G[H_{1},\ldots,H_{n}]$ is a spanning subgraph of $J_{i}^{j}$, for all
$1\leq i\leq n,\ 1\leq j\leq k_{i}$, it is enough to check that if $v$ and $w$
are not adjacent in $G[H_{1},\ldots,H_{n}]$, then they are not adjacent in
$J_{i}^{j}$ for some $i\in\\{1,2,\ldots,n\\}$ and
$j\in\\{1,2,\ldots,k_{i}\\}$. Let $vw\notin E(G[H_{1},\ldots,H_{n}])$. First,
if $v,w\in V(H_{i})$, for some $i\in\\{1,2,\ldots,n\\}$, then they are not
adjacent in $I_{i}^{j}$ for some $j\in\\{1,2,\ldots,k_{i}\\}$ and hence they
are not adjacent in $J_{i}^{j}$. Next, if $v\in V(H_{i})$ and $w\in V(H_{j})$
for $i\neq j$, then $u_{i}$ and $u_{j}$ are not adjacent in $G$ and hence $v$
and $w$ are not adjacent in $J_{i}^{j}$. Hence the Claim 2.
From Claims 1, 2 and by Theorem 1.1, the result follows. ∎

The following corollaries are the immediate consequence of Theorem 3.1.

######  Corollary 3.2.

If $G$ is a graph and $G_{r}$ is the reduced graph of $G$, then
$box(G)\leq|V(G_{r})|$.

######  Corollary 3.3.

If $G=K_{n_{1},n_{2},\ldots,n_{k}}$ (where $n_{i}\geq 2$) is a complete
$k$-partite graph, then $box(G)=k$.

######  Theorem 3.4.

Let $G$ be a graph with $V(G)=\\{u_{1},u_{2},\ldots,u_{n}\\}$ such that
$\langle\\{u_{1},u_{2},\ldots,u_{\ell}\\}\rangle$ is a clique in $G$ and
$\ell<n$. If $H_{1},H_{2},\ldots,H_{\ell}$ are pairwise disjoint complete
graphs and $H_{\ell+1},\ldots,H_{n}$ are pairwise disjoint graphs, then
$box(G[H_{1},H_{2},\ldots,H_{n}])\leq\mathop{\sum}\limits_{i=\ell+1}^{n}box(H_{i})$.

###### Proof.

For $\ell+1\leq i\leq n$, let $box(H_{i})=k_{i}$. By Theorem 1.1, there exist
interval graphs $I_{i}^{1},\ldots,I_{i}^{k_{i}}$ with vertex set $V(H_{i})$
such that $H_{i}=\mathop{\bigcap}\limits_{j=1}^{k_{i}}I_{i}^{j}$. For
$\ell+1\leq i\leq n$ and $1\leq j\leq k_{i}$, let $J_{i}^{j}$ be a graph as
defined in Theorem 3.1. Then we can prove that $J_{i}^{j}$ is an interval
graph and $G[H_{1},H_{2},\ldots,H_{n}]$ is a spanning subgraph of $J_{i}^{j}$
(the proof is same as in Theorem 3.1). We have to prove that
$E(G[H_{1},\ldots,H_{n}])=\mathop{\bigcap}\limits_{i=\ell+1}^{n}\mathop{\bigcap}\limits_{j=1}^{k_{i}}E(J_{i}^{j})$.

It is enough to prove that, if $xy\notin E(G[H_{1},H_{2},\ldots,H_{n}])$, then
there exist, $\ell+1\leq i\leq n$ and $1\leq r\leq k_{i}$ such that $xy\notin
E(J_{i}^{r})$.

If $x\in V(H_{i})$ and $y\in V(H_{j})$ for $\ell+1\leq i\leq j\leq n$, then by
the same proof as in Theorem 3.1, $xy\notin E(J_{i}^{r})$, for some
$\ell+1\leq i\leq n$ and $1\leq r\leq k_{i}$.

The only other possibility is: $x\in V(H_{i})$ for some $1\leq i\leq\ell$ and
$y\in V(H_{j})$, for some $\ell+1\leq j\leq n$. In this case, $u_{i}$ and
$u_{j}$ are not adjacent in $G$ and hence not adjacent in $R_{j}$ (see Theorem
3.1) and hence by the definition of $J_{j}^{r}$, $xy\notin J_{j}^{r}$, for all
$1\leq r\leq k_{j}$. ∎

##  4\. Boxicity of Zero-Divisor Graph of a ring

In this section, we obtain a bound for boxicity of the zero-divisor graph of a
ring.

Let $R$ be a ring. By Theorem 1.4, $\Gamma(R)\cong\Gamma_{E}(R)[\left\langle
C_{x_{1}}\right\rangle,\left\langle C_{x_{2}}\right\rangle,\ldots,\left\langle
C_{x_{k}}\right\rangle]$, where $|V(\Gamma_{E}(R))|=k$.

######  Theorem 4.1.

If the subgraph graph induced by $\\{x_{1},x_{2},\ldots,x_{r}\\}$ is a clique
in $\Gamma_{E}(R)$ with $x^{2}=0$, for $1\leq i\leq r$, then
$box(\Gamma(R))\leq k-r$, where $k=|V(\Gamma_{E}(R))|$.

###### Proof.

By Theorem 1.4, for $1\leq i\leq k$, $\left\langle C_{x_{i}}\right\rangle$ is
a complete graph or a totally disconnected graph. Then $box(\left\langle
C_{x_{i}}\right\rangle)=1$, for $1\leq i\leq k$. By Theorem 3.4,
$box(\Gamma(R))\leq k-r$. ∎

It is proved in [5] that if $R$ is a non-zero reduced ring, then there exists
positive integer $k$ such that the compressed zero-divisor graph
$\Gamma_{E}(R)$ of $R$ is isomorphic to $\Gamma(R^{\prime})$, that is
$\Gamma_{E}(R)\cong\Gamma(R^{\prime})$, where
$R^{\prime}=\mathbb{Z}_{2}^{k}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\ldots\times\mathbb{Z}_{2}$
($k$-copies). It is proved that in [8], $\chi(\Gamma(R))=k=\omega(\Gamma(R))$.

######  Theorem 4.2.

If $R$ is a non-zero reduced ring, then $\chi(\Gamma(R))\leq
box(\Gamma(R))\leq 2^{\chi(\Gamma(R))}-2$.

###### Proof.

By Theorem 1.4,
$\Gamma(R)\cong\Gamma(\mathbb{Z}_{2}^{k})[C_{x_{1}},\ldots,C_{x_{{2^{k}-2}}}]$.
Since $R$ is reduced, $x_{i}^{2}\neq 0$, for $1\leq i\leq 2^{k}-2$. Therefore,
$\left\langle C_{x_{i}}\right\rangle$’s are complement of complete graphs.
Hence, by Theorem 3.1, $box(\Gamma(R))\leq 2^{k}-2$ (because $box(\left\langle
C_{x_{i}}\right\rangle)=1$ and $|V(\mathbb{Z}_{2}^{k})|=2^{k}-2$).

Next we prove the lower bound. For $1\leq i\leq k$, let $e_{i}$ be the element
of $R^{\prime}$ with the $i^{th}$-coordinate is one and other coordinate are
zero. Clearly $\\{e_{i}\ |\ 1\leq i\leq k\\}$ forms a clique in $\Gamma(R)$
and $e_{i}^{2}\neq 0$, for $1\leq i\leq k$. Thus, $\left\langle
C_{e_{i}}\right\rangle$ is the complement of complete graph and hence by
Theorem 3.1, $k\leq box(\Gamma(R))$, because $box(\left\langle
C_{e_{i}}\right\rangle)=1$. ∎

##  5\. Boxicity of Zero-Divisor Graph of the ring $\mathbb{Z}_{N}$

In Section 5, we first prove that the chromatic number and the clique number
of $\Gamma_{E}(\mathbb{Z}_{N})$ are same for any positive integer $N$.

######  Lemma 5.1.

If
$N=\prod\limits_{i=1}^{a}p_{i}^{2n_{i}}\prod\limits_{j=1}^{b}q_{j}^{2m_{j}+1}$,
where $p_{i}$’s and $q_{j}$’s are distinct prime numbers, then
$\omega(\Gamma_{E}(\mathbb{Z}_{N}))=\chi(\Gamma_{E}(\mathbb{Z}_{N}))=\mathop{\prod}\limits_{i=1}^{a}(n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(m_{j}+1)+b-1.$

###### Proof.

Let $S=\\{d\in V(\Gamma_{E}(\mathbb{Z}_{N})):d\mid N\mid d^{2}\\}$. Then,
clearly $S$ is a clique in $\Gamma_{E}(\mathbb{Z}_{N})$.
Claim. $|S|=\prod\limits_{i=1}^{a}(n_{i}+1)\prod\limits_{j=1}^{b}(m_{j}+1)-1$.
Let $d$ be a positive integer such that

(6) $d\mid N\mid d^{2}$

Since $d\mid N$, we have $d$ is of the form
$\prod\limits_{i=1}^{a}p_{i}^{\alpha_{i}}\prod\limits_{j=1}^{b}q_{j}^{\beta_{j}}$,
where $0\leq\alpha_{i}\leq 2n_{i}$, $\forall\ i$ and $0\leq\beta_{j}\leq
2m_{j}+1$, $\forall\ j$. Since $\frac{N}{d}\mid d$, we have

$2n_{i}-\alpha_{i}\leq\alpha_{i}\mbox{ and }2m_{j}+1-\beta_{j}\leq\beta_{j}$

for all $1\leq i\leq a$ and $1\leq j\leq b$. Thus for all $1\leq i\leq a$,
$1\leq j\leq b$, we have

$n_{i}\leq\alpha_{i}\leq 2n_{i}\mbox{ and }m_{j}+1\leq\beta_{j}\leq 2m_{j}+1.$

Therefore, the number of positive integers $d$ satisfying the condition (6) is
$\prod\limits_{i=1}^{a}(n_{i}+1)\prod\limits_{j=1}^{b}(m_{j}+1)$. As the set
of all positive integers satisfying the condition (6) is $S\cup\\{N\\}$, we
get that

$|S|=\prod\limits_{i=1}^{a}(n_{i}+1)\prod\limits_{j=1}^{b}(m_{j}+1)-1.$

Hence the claim.

For $1\leq\eta\leq b$, we let

(7) $\xi_{\eta}=\prod\limits_{i=1}^{a}p_{i}^{2n_{i}}\prod\limits_{j=1\atop
j\neq\eta}^{b}q_{j}^{2m_{j}+1}\times q_{\eta}^{m_{\eta}}.$

Then $\xi_{\eta}\notin S$ and $T=S\cup\\{\xi_{\eta}:1\leq\eta\leq b\\}$ forms
a clique in $\Gamma_{E}(\mathbb{Z}_{N})$. Note that

(8)
$|T|=\prod\limits_{i=1}^{a}(n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(m_{j}+1)+b-1.$

Hence $\omega(\Gamma_{E}(\mathbb{Z}_{N}))\geq|T|$.
Next, we prove that $\chi(\Gamma_{E}(\mathbb{Z}_{N}))\leq|T|$. Let

(9)
$d=\prod\limits_{i=1}^{a}p_{i}^{r_{i}}\prod\limits_{j=1}^{b}q_{j}^{s_{j}}\in
V(\Gamma_{E}(\mathbb{Z}_{N}))\setminus T.$

Then, there exists $i$ with $1\leq i\leq a$ such that $r_{i}<n_{i}$ or there
exists $j$ with $1\leq j\leq b$ such that $s_{j}\leq m_{j}$.

Case 1. $r_{i}<n_{i}$ for some $i$ with $1\leq i\leq a$.
If $\sigma(d)=\max\\{i:r_{i}<n_{i},1\leq i\leq a,\mbox{ where }r_{i}\mbox{ is
as in \eqref{dfact}}\\}$ and $\overline{d}=\prod\limits_{i=1\atop
i\neq\sigma(d)}^{a}p_{i}^{2n_{i}}\times
p_{\sigma(d)}^{n_{\sigma(d)}}\times\prod\limits_{j=1}^{b}q_{j}^{2m_{j}+1}$,
then $\overline{d}\in S\subset T$ and $N\nmid d\overline{d}$, as
$r_{\sigma(d)}+n_{\sigma(d)}<2n_{\sigma(d)}$. So $d$ and $\overline{d}$ are
not adjacent in $\Gamma_{E}(\mathbb{Z}_{N})$.

Case 2. $r_{i}\geq n_{i}$ for all $1\leq i\leq a$ and there exists $j$ with
$1\leq j\leq b$ such that $s_{j}\leq m_{j}$.
If $\ell(d)=\max\\{j:s_{j}\leq m_{j},1\leq j\leq b\mbox{ where }s_{j}\mbox{ is
as in \eqref{dfact}}\\}$, then $\xi_{\ell(d)}\in T\setminus S$ (see (7) for
the definition of $\xi_{\eta}$) and $N\nmid\xi_{\ell(d)}d$, as
$s_{\ell(d)}+m_{\ell(d)}<2m_{\ell(d)}+1$. Therefore, $d$ and $\xi_{\ell(d)}$
are not adjacent in $\Gamma_{E}(\mathbb{Z}_{N})$.

Let $\psi:T\to\left\\{1,2,\ldots,|T|\right\\}$ be a bijective function. We
extend the function $\psi$ over $V(\Gamma_{E}(\mathbb{Z}_{N}))$ as
$c:V(\Gamma_{E}(\mathbb{Z}_{N}))\to\left\\{1,2,\ldots,|T|\right\\}$ defined by

$c(d)=\begin{cases}\psi(d)&\mbox{ if }d\in T\\\ \psi(\overline{d})&\mbox{ if
}d\not\in T\mbox{ and }d\mbox{ as in {\bf Case 1}}\\\
\psi(\xi_{\ell(d)})&\mbox{ if }d\not\in T\mbox{ and }d\mbox{ as in {\bf Case
2}}\end{cases}$

Finally, we prove that $c$ is a proper coloring of
$\Gamma_{E}(\mathbb{Z}_{N})$. Let $d^{\prime},d^{\prime\prime}\in
V(\Gamma_{E}(\mathbb{Z}_{N}))$.

  * •

If $d^{\prime},d^{\prime\prime}\in T$, then they receive distinct colors.

  * •

If $d^{\prime},d^{\prime\prime}\in V(\Gamma_{E}(\mathbb{Z}_{N}))\backslash T$
and $d^{\prime}$ is under Case 1 and $d^{\prime\prime}$ is under case 2, then
they receive different colors, as $\overline{d}\in S$ and $\xi_{\ell(d)}\in
T\setminus S$.

  * •

If $d^{\prime},d^{\prime\prime}\in V(\Gamma_{E}(\mathbb{Z}_{N}))\backslash T$,
both $d^{\prime},d^{\prime\prime}$ are under Case 1, and
$c(d^{\prime})=c(d^{\prime\prime})$, we show that $d^{\prime}$ and
$d^{\prime\prime}$ are not adjacent in $\Gamma_{E}(\mathbb{Z}_{N})$. Indeed,
$\psi(\overline{d^{\prime}})=\psi(\overline{d^{\prime\prime}})$ for some
$\overline{d^{\prime}},\overline{d^{\prime\prime}}\in S$ corresponding to
$d^{\prime}$ and $d^{\prime\prime}$, respectively, as introduced in Case 1.
Since $\psi$ is injective, we have
$\overline{d^{\prime}}=\overline{d^{\prime\prime}}$ and hence
$\sigma(d^{\prime})=\sigma(d^{\prime\prime})=\sigma$, say. This implies that
$r_{\sigma}(d^{\prime})+r_{\sigma}(d^{\prime\prime})<2n_{\sigma}$, where
$r_{\sigma}(d^{\prime})$ and $r_{\sigma}(d^{\prime\prime})$ are the powers of
$p_{\sigma}$ in $d^{\prime}$ and $d^{\prime\prime}$, respectively. Thus,
$N\nmid d^{\prime}d^{\prime\prime}$, and hence $d^{\prime}$ and
$d^{\prime\prime}$ are not adjacent in $\Gamma_{E}(\mathbb{Z}_{N})$.

  * •

If $d^{\prime},d^{\prime\prime}\in V(\Gamma_{E}(\mathbb{Z}_{N}))\backslash T$,
both $d^{\prime},d^{\prime\prime}$ are under Case 2, and
$c(d^{\prime})=c(d^{\prime\prime})$, then $d^{\prime}$ and $d^{\prime\prime}$
are not adjacent in $\Gamma_{E}(\mathbb{Z}_{N})$, by a similar argument used
in the previous case.

  * •

If $d^{\prime}\in T$, $d^{\prime\prime}\in
V(\Gamma_{E}(\mathbb{Z}_{N}))\backslash T$, and
$c(d^{\prime})=c(d^{\prime\prime})$, then
$d^{\prime}=\overline{d^{\prime\prime}}$ or
$d^{\prime}=\xi_{\ell(d^{\prime\prime})}$. If
$d^{\prime}=\overline{d^{\prime\prime}}$, then $d^{\prime}$ and
$d^{\prime\prime}$ are not adjacent, as
$n_{\sigma(d^{\prime\prime})}+r_{\sigma(d^{\prime\prime})}(d^{\prime\prime})<2n_{\sigma(d^{\prime\prime})}$,
equivalently, $N\nmid d^{\prime}d^{\prime\prime}$. Similarly, we can prove
that $d^{\prime}$ and $d^{\prime\prime}$ are not adjacent whenever
$d^{\prime}=\xi_{\ell(d^{\prime\prime})}$.

Therefore, $\chi(\Gamma_{E}(\mathbb{Z}_{N}))\leq|T|$. Thus, the proof is
completed. ∎

As an application of Theorem 4.1 and Lemma 5.1, we have

######  Theorem 5.2.

If
$N=\prod\limits_{i=1}^{a}p_{i}^{2n_{i}}\prod\limits_{j=1}^{b}q_{j}^{2m_{j}+1}$,
where $p_{i}$’s and $q_{j}$’s are distinct prime numbers. Then
$box(\Gamma(\mathbb{Z}_{N}))\leq\left(\mathop{\prod}\limits_{i=1}^{a}(2n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(2m_{j}+2)\right)-\left(\mathop{\prod}\limits_{i=1}^{a}(n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(m_{j}+1)\right)-1.$

###### Proof.

Let $\Gamma_{E}(\mathbb{Z}_{N})$ be the compressed zero-divisor graph of
$\mathbb{Z}_{N}$. Then the vertex set $V(\Gamma_{E}(\mathbb{Z}_{N}))$ of
$\Gamma_{E}(\mathbb{Z}_{N})$ is the set of all proper divisors,
$d_{1},d_{2},\ldots,d_{k}$, of $N$. Hence
$|V(\Gamma_{E}(\mathbb{Z}_{N}))|=k=\mathop{\prod}\limits_{i=1}^{a}(2n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(2m_{j}+2)-2$.

Let $S=\\{d\in V(\Gamma_{E}(\mathbb{Z}_{N})):d\mid N\mid d^{2}\\}$. Then by
the Lemma 5.1,

(10)
$|S|=\mathop{\prod}\limits_{i=1}^{a}(n_{i}+1)\mathop{\prod}\limits_{j=1}^{b}(m_{j}+1)-1,$

$S$ is a clique in $\Gamma_{E}(\mathbb{Z}_{N})$ and $x^{2}=0$ for $x\in S$. By
Theorem 4.1, $box(\Gamma(\mathbb{Z}_{N}))\leq|V(\Gamma_{R}(R))|-|S|$. Hence
the result follows. ∎

Finally, we characterize a composite number $N$ for which the boxicity of the
zero-divisor graph $\Gamma(\mathbb{Z}_{N})$ is one. We can easily observe the
following: If a graph $G$ contains an induced cycle of length 4 or more, then
$box(G)\geq 2$, that is, $G$ is not an interval graph.

######  Theorem 5.3.

Let $N$ be any composite number. Then, $box(\Gamma(\mathbb{Z}_{N}))=1$ if and
only if either $N=p^{n}$ for some prime number $p$ and for some integer $n\geq
2$ or $N=2p$ for some odd prime number $p$.

###### Proof.

For a proper divisor $d$ of $N$, define
$A_{d}=\\{a\in\mathbb{Z}_{N}:gcd(N,a)=d)\\}$. It is clear that $A_{d_{i}}\cap
A_{d_{j}}=\emptyset$ whenever $d_{i}$ and $d_{j}$ are distinct proper divisors
of $N$ and
$\Gamma(\mathbb{Z}_{N})\cong\Gamma_{E}(\mathbb{Z}_{N})[A_{d_{1}},A_{d_{2}},\ldots,A_{d_{k}}]$,
where $d_{i}$’s are all proper divisors of $N$. First, let us assume that
$box(\Gamma(\mathbb{Z}_{N}))=1$.

Suppose $N=p_{1}^{n_{1}}p_{2}^{n_{2}}\ldots p_{s}^{n_{s}}$, where $p_{i}$’s
are distinct primes.

Case 1. $n_{i}\geq 2$ for some $i$ with $1\leq i\leq s$.

We assume that $n_{1}\geq 2$. If $s\geq 2$, then the set of vertices

$\\{(p_{1}^{n_{1}-1}p_{2}^{n_{2}}\ldots
p_{s}^{n_{s}}),(p_{1}^{n_{1}}p_{3}^{n_{3}}\ldots
p_{s}^{n_{s}}),(p_{2}^{n_{2}}\ldots p_{s}^{n_{s}}),p_{1}^{n_{1}}\\}$

in $\Gamma(\mathbb{Z}_{N})$ induces a cycle of length 4, a contradiction.
Therefore, $s=1$ and hence $N=p_{1}^{n_{1}}$.

Case 2. $n_{i}=1$, for all $1\leq i\leq s$.

Since $N$ is composite number, we have $s\geq 2$. Suppose that there are two
odd prime factors for $N$, say $p_{1}\geq 3$ and $p_{2}\geq 3$, then
$K_{2}[\langle A_{p_{1}p_{3}p_{4}\ldots p_{s}}\rangle,\langle
A_{p_{2}p_{3}\ldots p_{s}}\rangle]=\langle A_{p_{1}p_{3}p_{4}\ldots
p_{s}}\rangle\vee\langle A_{p_{2}p_{3}\ldots p_{s}}\rangle$ is an induced
subgraph of $\Gamma(\mathbb{Z}_{N})$. Since $A_{p_{1}p_{3}p_{4}\ldots p_{s}}$
and $A_{p_{2}p_{3}\ldots p_{s}}$ are independent subsets of
$\Gamma(\mathbb{Z}_{N})$, the set $\\{(p_{1}p_{3}p_{4}\ldots
p_{s}),2(p_{1}p_{3}p_{4}\ldots p_{s}),(p_{2}p_{3}\ldots p_{s}),$
$2(p_{2}p_{3}\ldots p_{s})\\}$ of vertices of $\Gamma(\mathbb{Z}_{N})$ induces
a cycle of length $4$, (because $|A_{p_{1}p_{3}p_{4}\ldots p_{s}}|\geq 2$ and
$|A_{p_{2}p_{3}\ldots p_{s}}|\geq 2$), a contradiction. So, $N=2p$ for some
odd prime $p$.

Conversely, let us assume that either $N=p^{n}$ for some prime number $p$ and
for some integer $n\geq 2$ or $N=2p$ for some odd prime number $p$. If $N=2p$,
where $p$ is a prime, then $\Gamma(\mathbb{Z}_{N})\cong K_{2}$ and hence
$box(\Gamma(\mathbb{Z}_{N}))=1$. Now let us assume that $N=p^{n}$, where $p$
is a prime and $n\geq 2$. When $n=2$, $\Gamma(\mathbb{Z}_{p^{2}})\cong
K_{p-1}$ and hence $box(\Gamma(\mathbb{Z}_{p^{2}}))=1$. When $n=3$,
$\Gamma(\mathbb{Z}_{p}^{3})\cong\Gamma_{E}(\mathbb{Z}_{p^{3}})[\left\langle
A_{p}\right\rangle,\left\langle A_{p^{2}}\right\rangle]$, where $A_{p}$ is an
independent set, $\left\langle A_{p^{2}}\right\rangle$ is complete and every
vertex of $A_{p}$ is adjacent to every vertex of $A_{p^{2}}$. Let
$A_{p}=\\{v_{j}:1\leq j\leq|A_{p}|\\}$. We define an interval representation
$f$ of $\Gamma(\mathbb{Z}_{p^{3}})$. For $x\in A_{p^{2}}$, define $f(x)=[0,1]$
and for $v_{j}\in A_{p}$, define $f(v_{j})=\\{\frac{1}{j}\\}$. One can easily
verify that $f$ is an interval representation of $\Gamma(\mathbb{Z}_{p^{3}})$
and therefore $box(\Gamma(\mathbb{Z}_{N}))=1$.

When $n\geq 4$,
$\Gamma(\mathbb{Z}_{N})=\Gamma_{E}(\mathbb{Z}_{N})[\left\langle
A_{p}\right\rangle,\left\langle A_{p^{2}}\right\rangle,\ldots,\left\langle
A_{p^{n-1}}\right\rangle]$. For $1\leq i\leq n-1$, let $A_{p^{i}}=\\{v_{i,j}\
:1\leq j\leq|A_{p^{i}}|\\}$. Here one can easily observe the following.

  * •

If $n$ is even, then we see that the following.
(i) $A_{p^{i}}$ is an independent set if and only if
$i\in\\{1,2,\ldots,\frac{n}{2}-1\\}$;
(ii) $\left\langle A_{p^{j}}\right\rangle$ is complete if and only if
$j\in\\{\frac{n}{2},\ldots,n-1\\}$;
(iii) For $i\in\\{1,2,\ldots,\frac{n}{2}-1\\}$, every vertex of $A_{p^{i}}$ is
adjacent to every vertex of $A_{p^{n-j}}$ if and only if
$j\in\\{1,2,\ldots,i\\}$;
(iv) the set $\mathop{\bigcup}\limits_{j=1}^{\frac{n}{2}-1}A_{p^{j}}$ is
independent and
$\left\langle\mathop{\bigcup}\limits_{j=\frac{n}{2}}^{n-1}A_{p^{j}}\right\rangle$
is complete.

  * •

If $n$ is odd, then we see that the following.
(i) $A_{p^{i}}$ is an independent set if and only if
$i\in\\{1,2,\ldots,\left\lfloor\frac{n}{2}\right\rfloor\\}$;
(ii) $\left\langle A_{p^{j}}\right\rangle$ is complete if and only if
$j\in\\{\left\lceil\frac{n}{2}\right\rceil,\ldots,n-1\\}$;
(iii) For $i\in\\{1,2,\ldots,\left\lfloor\frac{n}{2}\right\rfloor\\}$, every
vertex of $A_{p^{i}}$ is adjacent to every vertex of $A_{p^{n-j}}$ if and only
if $j\in\\{1,2,\ldots,i\\}$;
(iv) the set
$\mathop{\bigcup}\limits_{j=1}^{\left\lfloor\frac{n}{2}\right\rfloor}A_{p^{j}}$
is an independent set and
$\left\langle\mathop{\bigcup}\limits_{j=\left\lceil\frac{n}{2}\right\rceil}^{n-1}A_{p^{j}}\right\rangle$
is complete.

We now construct an interval representation $f$ of $\Gamma(\mathbb{Z}_{N})$ as
follows.

For $1\leq i\leq\left\lfloor\frac{n}{2}\right\rfloor-1$, define

$f(v_{n-i,j})=\left[0,\left\lfloor\frac{n}{2}\right\rfloor-i+1\right],\mbox{
for every }j\in\\{1,2,\ldots,|A_{p^{n-i}}|\\}.$

For $\left\lceil\frac{n}{2}\right\rceil+1\leq i\leq n-1$, let
$x_{n-i,1},x_{n-i,2},\ldots,x_{n-i,|A_{p^{n-i}}|}$ be distinct points in
$\left(i-\left\lceil\frac{n}{2}\right\rceil,i-\left\lceil\frac{n}{2}\right\rceil+1\right)$.
Define

$f(v_{n-i,j})=\\{x_{n-i,j}\\},\mbox{ for every
}j\in\\{1,2,\ldots,|A_{p^{n-i}}|\\}.$

If $n$ is even, define

$f\big{(}v_{\frac{n}{2},j}\big{)}=[0,1],\mbox{ for every
}j\in\\{1,2,\ldots,|A_{p^{\frac{n}{2}}}|\\}.$

If $n$ is odd, consider distinct points
$y_{\left\lfloor\frac{n}{2}\right\rfloor,1},y_{\left\lfloor\frac{n}{2}\right\rfloor,2},\ldots,y_{\left\lfloor\frac{n}{2}\right\rfloor,|A_{p^{\left\lfloor\frac{n}{2}\right\rfloor}}|}$
in $(0,1)$ and define

$f\Big{(}v_{\left\lfloor\frac{n}{2}\right\rfloor,j}\Big{)}=\Big{\\{}y_{\left\lfloor\frac{n}{2}\right\rfloor,j}\Big{\\}},\mbox{
for every
}j\in\Big{\\{}1,2,\ldots,|A_{p^{\left\lfloor\frac{n}{2}\right\rfloor}}|\Big{\\}}$

and

$f\Big{(}v_{\left\lceil\frac{n}{2}\right\rceil,j}\Big{)}=[0,1],\mbox{ for
every }j\in\\{1,2,\ldots,|A_{p^{\left\lceil\frac{n}{2}\right\rceil}}|\\}.$

One can verify that $f$ is an interval representation of
$\Gamma(\mathbb{Z}_{N})$ and hence $box(\Gamma(\mathbb{Z}_{N}))=1$.

∎

### Acknowledgment

This research was supported by the University Grant Commissions Start-Up
Grant, Government of India grant No. F. 30-464/2019 (BSR) dated 27.03.2019.

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# CWISEP J193518.59–154620.3: An Extremely Cold Brown Dwarf in the Solar
Neighborhood Discovered with CatWISE

Federico Marocco NASA Postdoctoral Program Fellow  Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109,
USA  IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA
91125, USA  Dan Caselden Gigamon Applied Threat Research, 619 Western Avenue,
Suite 200, Seattle, WA 98104, USA  Aaron M. Meisner Hubble Fellow  National
Optical Astronomy Observatory, 950 N. Cherry Ave., Tucson, AZ 85719, USA  J.
Davy Kirkpatrick IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd.,
Pasadena, CA 91125, USA  Edward L. Wright Department of Physics and Astronomy,
UCLA, 430 Portola Plaza, Box 951547, Los Angeles, CA 90095-1547, USA
Jacqueline K. Faherty Department of Astrophysics, American Museum of Natural
History, Central Park West at 79th Street, NY 10024, USA  Christopher R.
Gelino  IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena,
CA 91125, USA  Peter R. M. Eisenhardt  Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  John W.
Fowler  230 Pacific St., Apt. 205, Santa Monica, CA 90405, USA  Michael C.
Cushing Department of Physics and Astronomy, University of Toledo, 2801 West
Bancroft St., Toledo, OH 43606, USA  Roc M. Cutri IPAC, Mail Code 100-22,
Caltech, 1200 E. California Blvd., Pasadena, CA 91125, USA  Nelson Garcia
IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125,
USA  Thomas H. Jarrett Department of Astronomy, University of Cape Town,
Private Bag X3, Rondebosch, 7701, South Africa  Renata Koontz  University of
California, Riverside, 900 University Ave, Riverside, CA 92521, USA  Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr.,
Pasadena, CA 91109, USA  Amanda Mainzer  Jet Propulsion Laboratory, California
Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  Elijah
J. Marchese  University of California, Riverside, 900 University Ave,
Riverside, CA 92521, USA  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  Bahram Mobasher
University of California, Riverside, 900 University Ave, Riverside, CA 92521,
USA  David J. Schlegel Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA  Daniel Stern Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA  Harry I. Teplitz
IPAC, Mail Code 100-22, Caltech, 1200 E. California Blvd., Pasadena, CA 91125,
USA

(Received April 22, 2019; Revised June 11, 2019)

###### Abstract

We present the discovery of an extremely cold, nearby brown dwarf in the solar
neighborhood, found in the CatWISE catalog (Eisenhardt et al., in prep.).
Photometric follow-up with Spitzer reveals that the object, CWISEP
J193518.59–154620.3, has ch1–ch2 = 3.24$\pm$0.31 mag, making it one of the
reddest brown dwarfs known. Using the Spitzer photometry and the polynomial
relations from Kirkpatrick et al. (2019) we estimate an effective temperature
in the $\sim$270–360 K range, and a distance estimate in the 5.6–10.9 pc
range. We combined the WISE, NEOWISE, and Spitzer data to measure a proper
motion of $\mu_{\alpha}\cos\delta=337\pm 69$ mas yr-1, $\mu_{\delta}=-50\pm
97$ mas yr-1, which implies a relatively low tangential velocity in the range
7–22 km s-1.

brown dwarfs – infrared: stars – proper motions – solar neighborhood

††journal: ApJ

##  1 Introduction

The census of objects in the solar neighborhood has been growing steadily in
recent years (Henry et al., 2018). The advent of large-area optical and near-
infrared surveys (e.g. 2MASS, Skrutskie et al. 2006; SDSS, York et al. 2000;
UKIDSS, Lawrence et al. 2007; VHS, McMahon et al. 2013; PanSTARRS, Chambers et
al. 2016; AllWISE, Cutri et al. 2013), and the recent Gaia second data release
(Gaia Collaboration et al., 2018), have given us the opportunity to identify
previously overlooked members of the 20 pc sample.

Despite recent discoveries of nearby ultracool dwarfs (Scholz & Bell, 2018;
Faherty et al., 2018; Cushing et al., 2018; Mamajek et al., 2018), the census
of the coldest, lowest mass constituents of the solar neighborhood remains
largely incomplete. Kirkpatrick et al. (2019) found that the completeness
limit for the T and Y dwarfs sample steeply declines as a function of
effective temperature ($T_{\rm eff}$), from 19 pc in the 900–1050 K interval,
down to 8 pc in the 300–450 K interval. At even lower $T_{\rm eff}$, only one
object has been so far identified, WISE J085510.83–071442.5 (hereafter WISE
J0855–0714, Luhman, 2014), at a distance of 2.3 pc.

The paucity of objects in this temperature regime has prevented us from
answering questions fundamental to astrophysics: how does star formation
create objects of extremely low mass, and with what efficiency? Whereas the
form of the mass function is well established for higher mass stars, it is far
less constrained for the lowest mass stars and brown dwarfs. Objects of the
lowest mass, including brown dwarfs, may, in fact, have several paths to
creation depending upon their birth environment. Studies of star formation
regions (e.g. Burgess et al., 2009) and nearby, young moving groups (Faherty
et al., 2016; Best et al., 2017) have shown that objects as low-mass as a few
Jupiter masses ($M_{\rm Jup}$) can form in isolation. Older, isolated field
objects with these masses therefore must exist, and will have had many Gyr to
cool, making them cold analogs to planets in exosolar systems. Although
establishing the diversity of low-mass star formation from cluster to cluster
is important, observing more of these frigid, free-floating objects in the
well-mixed field population will enable us to determine the frequency with
which low-mass objects are formed across the age of the Galaxy.

Preliminary results from Kirkpatrick et al. (2019) show that the cutoff mass
for star formation, if there is one, must be lower than 10 $M_{\rm Jup}$ since
the model predictions assuming such a cutoff are already underpredicting the
number of objects found. In order to test cutoffs of lower mass, we require
more objects of extremely low temperature, akin to WISE J0855–0714.

As its name suggests, WISE J0855–0714 was discovered using data from WISE
(Wright et al., 2010), whose $W1$ (3.4 $\mu$m) and $W2$ (4.6 $\mu$m) bands are
ideally placed to identify extremely cold brown dwarfs with their red $W1-W2$
colors (in contrast to the blue $W1-W2$ colors of stars) because the $W2$ band
measures the peak of the spectral energy distribution while the $W1$ band lies
in a region of strong methane absorption (Burrows et al., 1997). WISE
J0855–0714 is relatively bright in $W2$ (13.89$\pm$0.05 mag, Luhman, 2014),
and so, despite the “statistics of one”, Wright et al. (2014) estimated the
68% confidence range for the number of “0855-like” objects in the existing
AllWISE dataset to be 4–35, with a median of 15.

Finding more of these hidden solar neighbors is one of the goals of CatWISE, a
NASA Astrophysics Data Analysis Program (ADAP) funded project combining data
from the 2010 to 2016 phases of the WISE mission, to generate an all-sky
photometric and astrometric catalog (Eisenhardt et al., in prep.).

Here we present CWISEP J193518.59–154620.3 (hereafter CWISEP J1935–1546), an
extremely cold brown dwarf at $\sim$8 pc discovered in the preliminary CatWISE
catalog111CWISEP is the official designation for sources identified in the
CatWISE Preliminary catalog, see Eisenhardt et al., in prep.. Its $W1-W2$
color, and follow-up Spitzer photometry, suggest CWISEP J1935–1546 has an
effective temperature comparable to that of WISE J0855–0714, in the 270–360 K
range, making it one of the coldest brown dwarfs identified so far.

In Section 2, we briefly describe the CatWISE data processing and the
preliminary catalog content; Section 3 details the machine-learning-based
procedure used to identify CWISEP J1935–1546; in Section 4 we present our
Spitzer follow-up photometry, and in Section 5 we combine the Spitzer data
with the WISE data to refine the motion measurement for this target. Finally,
in Section 6 we derive the basic properties for this cold new member of the
solar neighborhood.

##  2 CatWISE

CatWISE is an infrared photometric and astrometric catalog consisting of
900,849,014 sources over the entire sky selected from WISE and NEOWISE data
collected from 2010 to 2016 at $W1$ and $W2$.

CatWISE adapted the AllWISE pipeline to work on the coadded WISE and NEOWISE
images provided by unWISE (Meisner et al., 2018a, b). A full description of
CatWISE is provided in Eisenhardt et al. (in prep.), and of the AllWISE
pipeline in Cutri et al. (2013) and Kirkpatrick et al. (2014). Here we
summarize the steps relevant to the discovery of CWISEP J1935–1546.

Source detection for the preliminary CatWISE catalog was performed using MDET
(Marsh & Jarrett, 2012), which works simultaneously in $W1$ and $W2$. The
full-depth unWISE coadds (Meisner et al., 2018b) were resampled from
2048$\times$2048 (2.75′′/pixel) format to the 4095$\times$4095 (1.375′′/pixel)
format used by MDET for WISE source detection, using the Image Co-addition
with Optional Resolution Enhancement software (ICORE, Masci, 2013), and an
appropriate point spread function (PSF). The PSF interpolation kernel smooths
the images, providing a matched filter for optimal detection of isolated point
sources. The “std” unWISE images were used for uncertainties, as these provide
the standard deviation at each coadd pixel of the individual WISE exposures.
The detection threshold was set at SNR = 1.8, yielding a differential source
reliability of 50% based on deeper Spitzer data from the S-COSMOS program
(Sanders et al., 2007).

The WPHOT software package developed for the AllWISE pipeline (Cutri et al.,
2013), was adapted to perform source photometry and astrometry for CatWISE.
Two main changes are worth highlighting here:

(1) For AllWISE, WPHOT propagates each source position detected by MDET in the
coadded images to individual exposures, solving for the least-square best-fit
to the PSF, to determine source position and fluxes (hereafter “stationary
fit”). An alternative fit is also performed, allowing for linear motion of the
source through the individual exposures (hereafter “motion fit”). This
solution provides, along with motion, an alternative measurement of position,
propagated to a chosen reference epoch, as well as fluxes.

For a given inertial position in each sky coverage, the $\sim$12 exposures
which are combined in each unWISE epoch coadd (Meisner et al., 2018a) are
obtained within less than two days. As a result, the position of sources
beyond the solar system can be assumed to be fixed for each epoch. Therefore
CatWISE used unWISE epoch coadd images in place of individual exposures when
running WPHOT.

(2) The other significant modification to the AllWISE version of WPHOT
involves the treatment of the PSF asymmetry. The WISE PSF is asymmetric with
respect to the scanning direction (which is along lines of ecliptic
longitude). The scan direction is similar for all individual images in an
epoch coadd, and a given inertial position is scanned in opposite directions
every six months, i.e. in consecutive epochs (except very near an ecliptic
pole). Since WPHOT was not designed to use a time-dependent PSF, CatWISE chose
to measure source properties separately for the (typically four or more) epoch
coadds in each of the two scan directions, and then merge the two results.

The Preliminary CatWISE Catalog is available at catwise.github.io, and will
soon be available on the NASA/IPAC Infrared Science
Archive222https://irsa.ipac.caltech.edu/ as well. From comparison to Spitzer,
the signal-to-noise-ratio = 5 Vega magnitude limits are $W1=17.58$ mag and
$W2=16.43$ mag (cf. $W1=16.90$ mag and $W2=15.95$ mag for AllWISE). From
comparison to Gaia DR2 (Lindegren et al., 2018), CatWISE measures motions to a
1$\sigma$ accuracy of 100 mas yr-1 for sources $\sim$3 mag fainter than those
measured with a similar accuracy in AllWISE, and and achieves motion
measurement accuracy that is 10 times better at $W2=15$ mag (Eisenhardt et
al., in prep.), as expected given the longer time baseline afforded by the
combined data.

##  3 Target selection

CWISEP J1935–1546 was found as part of our larger effort to identify and
characterize very cold brown dwarfs using CatWISE. The search was conducted
using the Python package XGBoost333https://xgboost.readthedocs.io/en/latest/
(Chen & Guestrin, 2016), which implements machine learning algorithms under
the gradient boosting framework.

In general applications of supervised learning with XGBoost, one trains and
evaluates an XGBoost model with a set of previously classified samples. This
ground truth dataset is used throughout the development process to find a
model with effective features and hyperparameters. Our application followed
this general supervised learning paradigm.

We evaluated multiple XGBoost classifiers in target selection. CWISEP
J1935–1546 was selected by our classifier that targets faint and red objects.
More specifically, the classifier is restricted to training with and
classifying point sources with $W2>14$ mag and $W1-W2>1$ mag, or within
$3\sigma$ of those limits, therefore consistent with T or Y spectral type (see
e.g. Kirkpatrick et al., 2011).

To build our training set, since the prevalence of our target class (herein
“positive”) is so low, and our compensatory sample weights for the remainder
of the data set (herein “negative”) are also low by consequence, we manually
classified samples for the positive class versus randomly sampled for the
negative class. We took confirmed objects from the literature, and motion-
confirmed objects from our candidate lists, and cross-matched them against
CatWISE to obtain their CatWISE data. We removed/corrected mismatches until we
were confident that the remaining training set held an insignificant number of
mismatched training objects. This set, consisting of $\approx$200 objects,
became our positive class.

We carefully selected sample weights to achieve robust classification.
Firstly, we weighted samples of the positive and negative class as the inverse
proportion of the total number of objects in each class within the training
data. Consequently, samples belonging to the low population positive class
received higher weight, and those belonging to the high population negative
class received lower weight, so that

$\sum_{i=0}^{n_{neg}}w_{neg,i}=\sum_{i=0}^{n_{pos}}w_{pos,i}$ (1)

where $w_{neg,i}$, $w_{pos,i}$ are the weight for a single member of the
negative and positive class, respectively, and $n_{neg}$, $n_{pos}$ are the
total number of objects in the negative and positive class.

Next, we distributed weights within the positive class by $W2$ magnitude,
creating an even distribution of total weights per given 0.5 bin of $W2$
magnitude. That is, our training data contained fewer faint $W2$ magnitude
positive class members, so their weights were proportionally higher than
bright $W2$ magnitude positive class members. In practice

$\sum_{i=0}^{n_{pos,W2}}w_{pos,W2,i}=\sum_{i=0}^{n_{pos,W2+0.5}}w_{pos,W2+0.5,i}$
(2)

where $w_{pos,W2,i}$, $w_{pos,W2+0.5,i}$ are the weight for a single member of
the positive class in a given 0.5 magnitude bin (e.g. $14.5\leq W2<15.0$ mag)
and in the adjacent magnitude bin, respectively, while $n_{pos,W2}$,
$n_{pos,W2+0.5}$ are the total number of objects in the two bins in question.

Machine-learning classifiers are defined by two different sets of parameters –
model parameters and hyperparameters (also referred to as tuning parameters).
Model parameters are estimated by the machine learning algorithm itself, from
the data, as part of the learning process. Hyperparameters on the other hand
cannot be estimated directly from the data, as they regulate the learning
process itself (see e.g. Kuhn & Johnson, 2013). Examples of hyperparameters
are the k in k-nearest neighbor interpolation, or the learning rate for
training. Finding the optimal set of hyperparameters is itself a complex
problem, and several approaches have been adopted (see Claesen & De Moor,
2015, and references therein).

We searched with Scikit-learn’s randomized cross-validation function for
optimized hyperparameters (RandomizedSearchCV; Pedregosa et al., 2011). The
function takes in a model, the training set, a selection of hyperparameters,
and distributions from which to draw their guesses. It then picks
hyperparameter values from the provided distributions, and trains and tests
the model, searching for the values that optimize the model performance.
However, experimentally we found that a low learning rate of 0.0135, manually
enforced outside of the parameter search, lead to the greatest reduction of
the classification error rate. The classification error rate is defined as
$n_{\rm wrong}/n_{\rm tot}$, where $n_{\rm wrong}$ is the number of
misclassified objects, and $n_{\rm tot}$ is the total number of classified
objects (see e.g. Tan, 2018, Chapter 4.2).

After first training each XGBoost classifier with our initial training set, we
applied it to the entire CatWISE catalog, and selected the objects (usually
between 10,000 and 25,000) with the highest predicted probability membership
in the positive class. We then visually inspected each object, using available
optical, near- and mid-infrared images (taken from DSS, 2MASS, UKIDSS,
PanSTARRS, and AllWISE) and the online image blinking/visualization tool
WiseView444http://byw.tools/wiseview (Caselden et al., 2018). Objects
confirmed to be real, with $W1-W2$ color visually consistent with $W1-W2>1$
mag, and with visible motion (confirming they are nearby), were added to the
positive class. Common false positives included objects that were found to be
unflagged artifacts, variable sources leading to spurious motion measurements,
and partly blended objects with contaminated photometry and/or astrometry.

We then iterated by re-training the classifier on the full training data, and
applied the re-trained classifier to the entire catalog to select another
batch of high probability positive class entries. Periodically, we would
validate both the manually labelled and randomly sampled training data to
remove mislabelled objects. We would do this by performing various train-test
splits and visually inspecting negatively labelled entries that had the
highest probability (among such entries) of belonging to the positive class,
as well as the converse case.

The selection yielded an initial sample of 131 late-T and Y dwarf candidates.
After further visual inspection, we prioritized 32 with either no detection or
a marginal detection in W1 (hinting at an extremely low temperature) and
visible motion (hinting at their proximity). These are being followed-up
through our Spitzer campaign (see Section 4) to obtain ch1 ($3.6\,\mu$m) and
ch2 ($4.5\,\mu$m) photometry to confirm/refute their nature and estimate
effective temperature and photometric distance. CWISEP J1935–1546 is the
reddest among the objects followed-up so far, with $W2=15.926\pm 0.085$ mag,
$W1-W2=2.58\pm 0.37$ mag , and ch1–ch2 = 3.24$\pm$0.31 mag (see next section).

##  4 Spitzer follow-up

Spitzer observations were taken as part of program 14034 (Meisner, PI). Seven
exposures of 30 s were taken in each band, and these exposures were dithered
using a random dither pattern of medium scale. The number of individual
exposures was chosen so that we would obtain a 5$\sigma$ detection at ch1–ch2
= 2.75 mag.

Our target is very faint in the ch1 mosaic, and to measure it we had to lower
the SNR for detection from 5 (the default value in MOPEX/APEX) down to 2. For
the aperture photometry, we used an aperture with a radius of 4 pixels
(aperture1 in the MOPEX output files) and a sky annulus with a 24-to-40-pixels
radius. For the PRF-fit photometry, we used a set of warm PRFs built by Jim
Ingalls (see Kirkpatrick et al. 2019), which are very similar to the warm
mission PRFs developed by Hora et al. (2012), that are available on the IRSA
website555See
http://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/calibrationfiles/.. For
the aperture photometry, the resulting raw fluxes were multiplied by the
aperture corrections recommended in Table 4.7 of the IRAC Instrument Handbook
– 1.208 for ch1 and 1.221 for ch2 – to obtain the flux in units of $\mu$Jy;
for PRF-fit photometry, the resulting raw fluxes were divided by the
correction factor recommended in Table C.1 of the IRAC Instrument Handbook –
1.021 for ch1 and 1.012 for ch2. These aperture and PRF-fit fluxes were then
converted from $\mu$Jy to magnitudes using the flux zero points in the
Handbook’s Table 4.1 (280.9$\pm$4.1 Jy in ch1 and 179.7$\pm$2.6 Jy in ch2),
propagating the uncertainty in zero point and flux into the final measurement
error. This final photometry is given in Table 1. The ch1 detection reported
here corresponds to a SNR of 3.6.

CWISEP J1935–1546 has ch1–ch2 = 3.24$\pm$0.31 mag (PRF; the aperture color is
3.46$\pm$0.26), overlapping with WISE J0855–0714 (3.55$\pm$0.07 mag), and
similar to the second reddest brown dwarf known, WISE J035000.32-565830.2
(3.25$\pm$0.10 mag; Kirkpatrick et al., 2012). Figure 1 shows the unWISE $W1$
and $W2$ coadds, and our Spitzer ch1 and ch2 mosaics.

Table 1: CatWISE and Spitzer photometry for CWISEP J1935–1546. Band | Magnitude | Notes
---|---|---
CatWISE W1 | 18.509$\pm$0.359 | stationary fit (w1mpro)
CatWISE W2 | 15.926$\pm$0.085 | stationary fit (w2mpro)
CatWISE W1 | 18.534$\pm$0.396 | motion fit (w1mpro_pm)
CatWISE W2 | 15.852$\pm$0.079 | motion fit (w2mpro_pm)
Spitzer ch1 | 19.089$\pm$0.262 | aperture
Spitzer ch2 | 15.633$\pm$0.018 | aperture
Spitzer ch1 | 18.892$\pm$0.314 | PRF fit
Spitzer ch2 | 15.647$\pm$0.023 | PRF fit
Figure 1: $1\times 1$ arcmin cutouts from the unWISE W1 and W2 epoch coadds
(top left and right), and the Spitzer ch1 and ch2 mosaic (bottom left and
right), centered around CWISEP J1935–1546. Red circles mark its position at
the two epochs shown.

##  5 Astrometry

The CWISEP J1935–1546 CatWISE measured motion is
$\mu_{\alpha}\cos\delta=400\pm 100$ mas yr-1, $\mu_{\delta}=-90\pm 120$ mas
yr-1. However, with the aid of our Spitzer follow-up observation, we have
obtained a better measurement of the target’s motion.

We first re-registered the unWISE epoch coadds to the Gaia astrometric frame.
We extracted sources from the individual epoch coadds using the CatWISE
pipeline. Because that was run on individual epochs, no “motion fit” was
possible, and therefore the positions used are those resulting from the
“stationary fit” (see Section 2).

We then selected a sample of bright reference stars to be used for re-
registration. We retained only stars with $\sigma_{\alpha}$,
$\sigma_{\delta}<0.1^{\prime\prime}$, $W1\,>\,8.1$ mag and $W2\,>\,6.7$ mag
(the saturation limits for WISE; Cutri et al., 2012). After these cuts were
applied, our initial re-registration set consisted of more than 10,000 stars
at each epoch.

We cross-matched our re-registration set to Gaia DR2. We used Gaia DR2
astrometry to correct the positions of all of the Gaia stars in the CatWISE
field to the CatWISE epoch in question, and matched them to the re-
registration set using a 2.75′′ radius (corresponding to one unWISE pixel),
retaining only the closest matching Gaia source for each re-registration star.
We typically found $\sim$700 Gaia stars at each epoch.

The re-registration stars were used to fit both a 6th order and a 12th order
transformation between each epoch coadd and Gaia DR2, using our own IDL code.
The fit included a $3\sigma$ clipping iteration, removing re-registration
stars whose re-registered position after the first fit iteration was off by
more than three times the formal errors of the fit from their Gaia DR2
position. The use of a 12th order transformation did not significantly reduce
the residuals of the fit, and therefore we adopted the 6th order
transformation.

To determine reliable uncertainties on the coordinates of CWISEP J1935–1546 at
each epoch, we examined the dispersion between the re-registered positions of
all stars in the field (without any restriction on their positional accuracy),
and their Gaia DR2 positions. We found that at $W2\sim 16$ mag (the brightness
of CWISEP J1935–1546) the formal uncertainties reported by WPHOT underestimate
by a factor of $\sim$1.6 the observed dispersion. We therefore multiply the
formal uncertainties by that factor. CatWISE positions and our adopted
uncertainties are listed in Table 2.

For the re-registration of the Spitzer ch2 mosaic, and the measurement of
CWISEP J1935–1546 at that epoch, we adopted the same method described in
Kirkpatrick et al. (2019).

Finally, we performed both a linear fit and a 5-parameter astrometric fit
(position + proper motion + parallax motion) to the CatWISE and Spitzer ch2
positions of CWISEP J1935–1546 as a function of time (listed in Table 2). The
target falls below the signal-to-noise-ratio threshold for detection by the
CatWISE pipeline (SNR = 1.8) in two of the nine WISE plus NEOWISE epochs.

Table 2: Measured positions for CWISEP J1935–1546. The 2019 position was obtained through our Spitzer follow-up. The target is undetected in two of the nine unWISE epoch coadds. R.A. | $\sigma_{\rm R.A.}$ | Dec. | $\sigma_{\rm Dec.}$ | Epoch
---|---|---|---|---
(deg) | (arcsec) | (deg) | (arcsec) |
293.8267517 | 0.9 | -15.7722349 | 1.4 | 2010.2842
293.8270569 | 1.0 | -15.7723198 | 1.3 | 2010.7764
293.8275452 | 1.2 | -15.7720318 | 1.1 | 2015.2857
293.8275146 | 1.1 | -15.7723351 | 1.4 | 2015.7695
293.8275452 | 1.1 | -15.7723131 | 1.3 | 2016.7525
293.8276062 | 1.4 | -15.7726307 | 1.4 | 2017.2799
293.8276978 | 1.1 | -15.7723455 | 1.2 | 2017.7436
293.8277930 | 0.01 | -15.7723462 | 0.01 | 2019.0267

The linear fit yielded $\mu_{\alpha}\cos\delta=337\pm 69$ mas yr-1,
$\mu_{\delta}=-50\pm 97$ mas yr-1, while the 5-parameter astrometric fit
yielded $\mu_{\alpha}\cos\delta=341\pm 90$ mas yr-1, $\mu_{\delta}=-36\pm 113$
mas yr-1, and a trigonometric parallax of $-100\pm 440$ mas. The negative
parallax is clearly unphysical and not statistically significant. We therefore
adopt the values from the linear fit. The results of the linear fit are
presented in Figure 2.

Figure 2: A linear fit to the coordinates of CWISEP J1935–1546 as a function
of time. R.A. (red diamonds), Dec. (blue triangles), and time are relative to
their variance-weighted mean.

##  6 Analysis

The ch1–ch2 color for CWISEP J1935–1546 is comparable to that of the coldest
brown dwarf known, WISE J0855–0714. In Figure 3 we show $T_{\rm eff}$ as a
function of Spitzer ch1–ch2 color for a sample of known late-T and Y dwarfs
from the literature (see Kirkpatrick et al., 2019, and references therein).

Figure 3: $T_{\rm eff}$ as a function of Spitzer ch1–ch2 colors for nearby
late-T and Y dwarfs. Black points are objects taken from Kirkpatrick et al.
(2019, Table 8). The red dashed lines encompass the $1\sigma$ color range for
CWISEP J1935–1546. Overplotted in blue is the polynomial relation presented in
Kirkpatrick et al. (2019).

We can use our Spitzer photometry and the recent polynomial relations
presented in Kirkpatrick et al. (2019) to derive an effective temperature
estimate and photometric distance for CWISEP J1935–1546. Using the ch1–ch2
color to absolute ch2 magnitude relation, we obtain a $1\sigma$ distance
estimate of 5.6–10.9 pc. Given this estimated distance and the proper motion
measured here, CWISEP J1935–1546 has an estimated tangential velocity in the
range 7–22 km s-1, consistent with the tangential velocity distribution for L,
T and Y dwarfs in the solar neighborhood from Smart et al. (2019) and
Kirkpatrick et al. (2019).

The ch1–ch2 colour to $T_{\rm eff}$ relation from Kirkpatrick et al. (2019)
indicates a $T_{\rm eff}$ in the range $\sim$270–360 K, which would make
CWISEP J1935–1546 one of the coldest brown dwarfs discovered so far (see
Figure 3).

Given the temperature derived above, we can estimate a mass for this object
using the BT-Settl models (Allard et al., 2012, 2013). If we assume CWISEP
J1935–1546 is a field object, with age in the $\sim$500 Myr – 13 Gyr range, it
would have a mass in the range 2–20 $M_{\rm Jup}$. We can narrow down the age
and mass range by taking into account the fact that the tangential velocity
estimated here is consistent with the population of nearby ultracool dwarfs,
whose age is in the range $\sim$1.5–6.5 Gyr (see e.g. Wang et al., 2018, and
references therein). In this age range, CWISEP J1935–1546 would have a mass
between 3 and 14 $M_{\rm Jup}$.

No other photometry is currently available for this object, as it is well
below the detection threshold for existing optical and near-infrared surveys.
The position of CWISEP J1935–1546 is covered by VHS and PanSTARRS, but the
target is undetected in both, as well as in the $W3$ and $W4$ images from
AllWISE. Given the $T_{\rm eff}$ and distance ranges estimated above, the
expected $H$ magnitude would be 23.7–25.1 mag, a depth prohibitive for most
ground-based facilities, particularly for spectroscopy. Spectroscopic
characterization for this extremely cold object will necessarily have to wait
for the launch of JWST.

The discovery of CWISEP J1935–1546 starts to bridge the existing gap between
known warmer Y dwarfs and the extremely cold WISE J0855–0714. CatWISE, as well
as “Backyard Worlds: Planet 9” (a NASA-funded citizen science project; Kuchner
et al., 2017) are now fully exploiting the potential of the WISE and NEOWISE
data set to uncover more of these frigid, free-floating planetary mass
objects. CWISEP J1935–1546 is part of a larger sample of discoveries by these
two highly complementary projects, and joint observing campaigns are now
underway to fully characterize this compelling population. By populating this
region of parameter space we can not only put strong observational constraints
on the mass function for extremely low mass objects, but also understand the
processes that shape such cold, planet-like atmospheres.

This research was partly carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with NASA. FM is
supported by an appointment to the NASA Postdoctoral Program at the Jet
Propulsion Laboratory, administered by Universities Space Research Association
under contract with NASA. AMM acknowledges support from Hubble Fellowship HST-
HF2-51415.001-A. CatWISE is funded by NASA under Proposal No. 16-ADAP16-0077
issued through the Astrophysics Data Analysis Program, and uses data from the
NASA-funded WISE and NEOWISE projects. This work is based in part on
observations made with the Spitzer Space Telescope, which is operated by the
Jet Propulsion Laboratory, California Institute of Technology under a contract
with NASA.

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