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% Title (or short title) and author name for the header
\runauth{
All unicyclic graphs of order $n$ with locating-chromatic number $n-3$
\hspace{2ex} $\arrowvert$\hspace{2ex}
E.T. Baskoro and Arfin}


\begin{document}

\begin{frontmatter}
% Write your paper title here
\papertitle{All unicyclic graphs of order $n$ with locating-chromatic number $n-3$}

%% use optional labels to link authors explicitly to addresses. The label may be more than one, use comma to separate

%\author[label1,label]{First Author${}^1$}

\author[1]{Edy Tri Baskoro}
\author[2]{Arfin}


\address[1]{\small Combinatorial Mathematics Research Group,
Faculty of Mathematics and Natural Sciences,\\
Institut Teknologi Bandung, Jalan Ganesa 10 Bandung 40132, Indonesia.}
\address[2]{Master Program in Mathematics for Teaching, \\ Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, \\ Jalan Ganesa 10 Bandung 40132, Indonesia
{}
%\address[label2]{Combinatorial Mathematics Research Group,
%Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 10 Bandung 40132, Indonesia.}
%\address[label3]{Combinatorial Mathematics Research Group,
%Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesa 10 Bandung 40132, Indonesia.

\vspace*{2.5ex}
 {\normalfont ebaskoro@math.itb.ac.id, yong\_arfin@yahoo.co.id}
 }

\begin{abstract}
Characterizing all graphs having a certain locating-chromatic number is not an easy task. In this paper, we are going to pay attention on finding all unicyclic graphs of order $n \;(\geq 6)$ and having locating-chromatic number $n-3$.

\let\thefootnote\relax\footnotetext{Received: xx xxxxx 20xx,\quad
  Accepted: xx xxxxx 20xx.\\[3ex]
  }
\end{abstract}

\begin{keyword}
% Separate keyword by \sep
locating-chromatic number, unicyclic, graph, characterization
% Write the classification number
Mathematics Subject Classification : 05C12
\end{keyword}

\end{frontmatter}

%% Main text
\section{Introduction}
	\label{sec:1}
	
	Let $G=(V,E)$ be a connected graph. For any two vertices $a$ and $b$ in $G$, define 
	the {\em distance} between $a$ and $b$, denoted by $d(a,b)$, is the length of a shortest 
	path connecting $a$ and $b$. The {\em distance} from a vertex $a$ to a set $S$ in $G$, denoted by 
	$d(a,S)$, is $\min\{d(a,x)\mid{x}\in{S}\}$.
	Let $\Pi=\{L_1,L_2,...,L_k\}$ be an ordered partition of $V(G)$ induced by a $k$-coloring $c$. 
	The \textit{color code} $c_\Pi (v)$ of a vertex $v$ of $G$ is defined as 
	\begin{center} 
		$c_\Pi (v)=(d(v,L_1 ),d(v,L_2 ),...,d(v,L_k ))$. 
	\end{center} 
	%where $d(v,L_i )=\min\{d(v,x)\mid{x}\in{L_i}\}$  for $1\leq i\leq k$. 
	If any two distinct vertices $u$ and $v$ of $G$ satisfy that $c_\Pi(u)\neq{c_\Pi(v)}$, then the coloring $c$ is called a \textit{locating-coloring} of $G$. The {\em locating-chromatic number} of $G$, denoted by $\chi_L(G)$, is the smallest integer $k$ such that $G$ has a locating-coloring with $k$ colors. 
	
	Chartrand \textit{et al.} \cite{Chartrand-36-02} introduced the notion of the locating-chromatic number of a graph. 
	They derived some bounds of the locating-chromatic number of a graph in terms of its order and diameter. The locating-chromatic numbers of some well-known graphs are also obtained, such as for paths, cycles, double stars, and complete multipartite graphs. The existence of a tree of order $n\geq{5}$ having locating-chromatic number $k$ for any $k\in\{3,4,...,n-2,n\}$ is also shown. In \cite{FM2019}, Furuya and Matsumoto have proposed an algorithm to estimate an upper bound for the locating-chromatic number of any tree. This bound depends on the number of leaves and the number of local end-branches in a tree. Recently, Assiyatun et al. \cite{ASB2019} proposed an improved algorithm for calculating the upper bound for the locating-chromatic number of any tree. The bound obtained is much better than the one of Furuya and Matsumoto.
	
All connected graphs of order $n$ and having locating-chromatic number $n$ have been completely characterised, i.e., complete multipartite graphs, see \cite{Chartrand-36-02}. For small locating-chromatic number, all connected graphs with locating-chromatic number 3 have been characterized, see \cite{Baskoro-2-13} and \cite{Asmi-12}. 
In particular for trees, Syofyan, \textit{et al.}. \cite{Syofyan-10-13} has found all trees of order $n$ with 
locating-chromatic number $t$, where $2\leq t<\frac{n}{2}$.
Furthermore, in \cite{Chartrand-269-03}, Chartrand \textit{et al.} characterized all connected graphs of order $n$ and having locating-chromatic number $n-1$. However, the problem on characterizing all connected graphs of order $n$ and having locating-chromatic number $n-2$ is still open. 
A graph is called {\em unicyclic} if it contains exactly one cycle.
Recently, Arfin and Baskoro \cite{Arfin-18} characterized all unicyclic graphs of order $n\geq5$ with locating-chromatic number $n-2$. Such graphs are presented in the following theorem. In this paper, we characterize all unicyclic graphs of order $n \;(\geq 6)$ with locating-chromatic number $n-3$.

	
\begin{thm}\cite{Arfin-18}\label{theo1.1}
	There are exactly 9 non-isomorphic unicyclic graphs of order $n \geq 5$, listed in Figure 1, with locating-chromatic number $n-2$.
\end{thm}
	
	\begin{figure}[thpb]\label{fig1}
	\begin{center}
		\includegraphics[scale=.20]{Fig1.png} %0.28
		\caption{All unicyclic graphs of order $n \geq 5$ with locating-chromatic number $n-2$.}
	\end{center}
	\end{figure}
		
\section{Basic Properties}\label{sec:2}
In this section, we give some basic properties of locating-chromatic number of graphs.	
Let $G(V,E)$ be a nonempty connected graph of order $n$. 
The \textit{degree} of vertex $v$ in $G$, denoted by $deg(v)$, is the number of vertices in $G$ that are adjacent to $v$. A vertex of degree one is called an \textit{end-vertex} or a \textit{leaf} of $G$. 
The \textit{external degree} of a vertex $v$ in $G$, denoted by $d^+(v)$, is the number of leaves adjacent to $v$. The \textit{maximum external degree} of a graph $G$ is $\max\{d^+(v) \mid v \in V(G) \}$ and denoted by $\Delta^+(G)$. The following observation and corrolary are natural.

	\begin{obs}\cite{Chartrand-36-02}
		Let $c$ be a locating-coloring in a connected graph $G$. 
		If $u$ and $v$ are distinct vertices of $G$ such that $d(u,w)=d(v,w)$ for all $w\in{V(G)\backslash\{u,v\}}$, then $c(u)\neq{c(v)}$. In particular, if $u$ and $v$ are nonadjacent vertices of $G$ such that $N(u)=N(v)$, then $c(u)\neq{c(v)}$. 
	\end{obs}

	\begin{cor}\cite{Chartrand-36-02}\label{cor2.1}
		If $G$ is a connected graph containing a vertex $v$ with $d^+(v)=p$, then $\chi_L(G)\geq{p+1}$. Furthermore, if $\Delta^+(G)=P$, then $\chi_L(G)\geq{P+1}$.
	\end{cor}

Furthermore, Chartrand, \textit{et al.} \cite{Chartrand-36-02} derived some bounds on the locating-chromatic number of a connected graph in relation with its order and diameter, as shown in the following theorem. 
	
	\begin{thm}\cite{Chartrand-36-02}\label{theo2.1}
		If $G$ is a graph of order $n \geq 3$ and $diam(G) \geq  2$, then
		\begin{center}
			$\log_{d+1} n \leq \chi_L (G) \leq n-diam(G)+2$
		\end{center}
	\end{thm}
		
\noindent Note that $diam(G)$ is the diameter of graph $G$.
As a direct consequence of theorem \ref{theo2.1}, we have the following corollaries.
		
	\begin{cor}\label{cor2.2}
		If $G$ is a graph of order $n\geq 6$ with locating-chromatic number $n-3$, 
		then $2\leq diam(G)\leq 5$.
	\end{cor}
	
	\begin{cor}\label{cor2.3}
		If $k$ is the length of a cycle in a unicyclic graph $G$ of order $n \;(\geq 6)$ with locating-chromatic number $n-3$, then $3\leq k\leq 11$.
	\end{cor}

A tree $T$ for which a vertex $v$ is distinguished is called a \textit{rooted tree} 
and the distinguished vertex is called a \textit{root} of the tree. A rooted tree will be considered to be \textit{leveled}, \textit{i.e.} level $0$ contains the root, $v$, of the tree, level $1$ consists of all vertices adjacent to $v$, etc. 
A rooted tree $T$ is called \textit{trivial} if it is of order 1, otherwise it is \textit{nontrivial}. 
Let $H$ be a unicyclic graph containing a cycle of length $k$. Then, the graph $H$ 
can be also considered as the graph obtained from $k$ rooted trees $T_i$ of roots 
$a_i (1\leq i\leq k)$ by connecting all these roots into a cycle $C_k$ such that:
	
	\begin{center}
		$V(H)=\bigcup\limits_{i=1}^{k} V(T_i)$ and $E(H)=\bigg(\bigcup\limits_{i=1}^{k} E(T_i)\bigg)\cup E(C_k)$.
	\end{center}
	
In this paper, we denote by $\mathbb{H}$ the set of all unicyclic graphs $H$ of order $n \geq 6$ with $\chi_L(H)=n-3$. Note that there is no such unicyclic graph $H$ of order $n \leq 5$ with $\chi_L(H)=n-3$.

\section{Maximum external degree}\label{sec:3}

In this section, we are going to show that every unicyclic graph $H$ of order $n\geq 8$ with $\chi_L(H)=n-3$ must have the maximum external degree $n-4$, namely  $\Delta^+(H)=n-4$. To do this, 
let us first consider the following lemma.

	\begin{lem} \label{lem3}
		If $H$ is a unicyclic graph of order $n\geq 8$ with $\Delta^+(H)=1$, then $\chi_L(H)\leq n-4$.
	\end{lem}

	\begin{proof}
		Let $H$ be a unicyclic graph of order $n\geq 8$ with $\Delta^+(H)=1$. 
		Let $k$ be the length of the unique cycle in $H$. Then, consider the following two cases.

		\textit{Case 1:} $3\leq k\leq7$. Consider any connected subgraph $I$ of $H$ of order $8$ and containing the unique cycle with $\Delta^+(I)=1$. Then, all these possible subgraphs $I$ for each $k$ are shown in Figures 2 and 3, along with their minimum locating-colorings.
%%
		\begin{figure}[!h]\label{fig2}
		\begin{center}
			\includegraphics[scale=.109]{Fig2.png} %0.15
			\caption{All the subgraphs $I$ of $k=3$ or $k=4$ along with their minimum locating-colorings.}
		\end{center}
		\end{figure}

		\begin{figure}[!h]\label{fig3}
		\begin{center}
			\includegraphics[scale=.06]{Fig3.png} %0.095
			\caption{All the subgraphs $I$ with $k=5,6$, or $k=7$ along with their minimum locating-colorings.}
		\end{center}
		\end{figure}
		%%
		It can be seen that every subgraph $I$ in Figures 2 and 3 has a minimum locating-coloring with either 3 or 4 colors. Now extend this coloring into $H$ by coloring all the remaining vertices in $H$ with new different colors. By this way, we obtain a locating-coloring in $H$ with at most $(n-8)+4=n-4$ colors. Therefore, $\chi_L (H)\leq n-4$.

		\textit{Case 2:} $k\geq8$. Now, consider the unique cycle $C_k$ in $H$ and let $V(C_k)=\{a_i\mid 1\leq i\leq k\}$. If $k$ is odd, then define a coloring $c:V(C_k)\rightarrow \{1,2,3\}$ with:
	
		\[
		c(a_i)=\left\{
		\begin{array}{ll}
		$1$, & \text{if $i=1$}\\
		$2$, & \text{if $i$ is even}\\
		$3$, & \text{if $i\geq3$ and $i$ is odd.}
		\end{array}
		\right.
		\]
	
		\noindent If $k$ is even, then define a coloring $c:V(C_k)\rightarrow \{1,2,3,4\}$ with:
	
		\[
		c(a_i)=\left\{
		\begin{array}{ll}
		$1$, & \text{if $i=1$}\\
		$2$, & \text{if $i=2$}\\
		$3$, & \text{if $i\geq3$ and $i$ is odd}\\
		$4$, & \text{if $i\geq4$ and $i$ is even.}
		\end{array}
		\right.
		\]
	
		\begin{figure}[!ht]\label{fig4}
		\begin{center}
			\includegraphics[scale=.11]{Fig4.png} %0.16
			\caption{A locating-coloring $c$ in the unique cycle.}
		\end{center}
		\end{figure}
	
		\noindent It is clear that $c$ is a locating-coloring in $C_k$. 
		Now, extend this coloring $c$ into $H$ by coloring all the remaining vertices in $H$ with 
		new different colors. Of course, this extended coloring is a locating-coloring in $H$. 
		Then, we obtain $\chi_L(H) \leq n-4$.
	\end{proof}

	\begin{thm}\label{teo3.1}
		If $H$ is a unicyclic graph of order $n\geq 8$ with $\chi_L(H)=n-3$, 
		then  $\Delta^+(H)=n-4$.
	\end{thm}

	\begin{proof}
		Let $H$ be a unicyclic graph of order $n\geq 8$ with $\chi_L(H)=n-3$. 
		If $\Delta^+(H) \geq n-3$, then by Corollary \ref{cor2.1} we have $\chi_L(H) \geq n-3+1=n-2$, a contradiction. Therefore, $\Delta^+(H) \leq n-4$. 
		Now, assume $\Delta^+(H) < n-4$. Let $x$ be a vertex with maximum external degree, \textit{i.e.}		$d^+(x)=\Delta^+(H) \leq n-5$. 
	
		If $\Delta^+(H)=0$, it follows that $H\cong C_n$ which means $\chi_L(H)=3$ for odd $n$ 
		or $4$ for even $n$, a contradiction.
		If $\Delta^+(H)=1$, then by Lemma \ref{lem3}, we have $\chi_L(H)\leq n-4$, a contradiction. 
		Therefore, $2\leq \Delta^+(H) \leq n-5$. 
		Let $u_1,u_2, \cdots, u_{\Delta^+(H)}$ be the leaves adjacent to $x$ in $H$.
		By Corollary \ref{cor2.1} the vertices $x, u_1, u_2, \cdots, u_{\Delta^+(H)}$ must be assigned with distinct colors, say $1, 2, \cdots, \Delta^+(H)+1$. Now, consider the remaining vertices other than $x$ and its leaves in $H$. Let $J$ be a subgraph induced by these remaining vertices, say $V(J)=\{v_1,v_2,\cdots, v_{n-\Delta^+(H)-1}\}$. Then, there are at least 5 vertices in $J$. Let $p$ and $q$ be two non-adjacent vertices in $J$ such that $d(p,w)\neq d(q,w)$ for some $w\in V(H)\backslash\{p,q\}$. Define a coloring such that $p$ and $q$ are assigned with the same color, and the other  $n-\Delta^+(H)-3$ remaining vertices in $J$ are assigned with distinct colors different from the colors of $p$ and $q$. Such a coloring of $H$ is a locating-coloring, hence $\chi_L(H)\leq \max\{\Delta^+(H)+1, n-\Delta^+(H)-2\} \leq n-4$, which is a contradiction. Therefore, $\Delta^+(H)=n-4$.
	\end{proof}
	
\section{Characterization}\label{sec:4}

	Let $H$ be a unicyclic graph of order $n\geq6$ with $\chi_L(H)=n-3$. In this section, we will characterize all graphs $H$.
		
	\begin{thm}\label{teo4.1}
		There are exactly three non-isomorphic unicyclic graphs $H$ of order $n \geq 8$ with  $\chi_L(H)=n-3$. 
	\end{thm}
	
		\begin{figure}[thpb]\label{fig5}
		\begin{center}
			\includegraphics[scale=.23]{Fig5.png} %0.33
			\caption{Three non-isomorphic unicyclic graphs $H$ of order $n$ and $\chi_L(H)=n-3$ with their minimum locating-colorings.}
		\end{center}
		\end{figure}
	
	\begin{proof}
		Let $H$ be a unicyclic graph of order $n \geq 8$ and $\chi_L(H)=n-3$.  By Theorem \ref{teo3.1}, we have $\Delta^+(H)=n-4$. Let $x$ be a vertex of $H$ with maximum external degree, \textit{i.e.} $d^+(x)=\Delta^+(H)=n-4$. Then, there are exactly three remaining vertices other than $x$ and its leaves. The connected subgraph induced by these three vertices together with $x$ will contain a unique cycle. Therefore, there are exactly three possible graphs $H$ up to isomorphism (see Figure 5). For the converse, by Corollary \ref{cor2.1}, we have that $\chi_L(H) \geq n-3$. Next, each of these three graphs has a locating-coloring with $n-3$ colors (see Figure 5), hence $\chi_L(H)\leq n-3$. Therefore, for each of these graphs $H$, we have $\chi_L(H)=n-3$.
	\end{proof}
	
To complete the characterization, we have to find all the unicyclic graphs $H$ of order $n\leq 7$ with the required locating-chromatic number. Our search will be based on the length of the unique cycle $C_k$ in $H$.
	
	\begin{thm}\label{teo4.2}
		There are exactly two non-isomorphic unicyclic graphs $H$ of order $n\leq7$ with $ \chi_L(H)=n-3$ containing $C_k$ for $k \geq 5$. 
	\end{thm}

	\begin{figure}[thpb]\label{fig6}
	\begin{center}
		\includegraphics[scale=.15]{Fig6.png} %0.21
		\caption{Graphs $C_6$, $G_1$, $G_2$, $G_3$, $G_4$, and $G_5$, each with its minimum locating-coloring.}
	\end{center}
	\end{figure}

	\begin{proof}
		Let $H$ be a unicyclic graph of order $n\leq 7$ with $\chi_L(H)=n-3$ and containing the cycle
		of length $k \geq 5$. Then, $k=5, 6$, or $7$.
		If $k=7$ then $H \cong C_7$ and $\chi_L(C_7)=3 \; (=n-4)$, a contradiction.
		If $k=$ 5 or 6, then $H$ must be isomorphic to $C_6$, $G_1$, $G_2$, $G_3$, $G_4$, or $G_5$ (see Figure 6). We can see that $G_1$ and $G_5$ are the only graphs having the required
		locating-chromatic number. 
	\end{proof}
	
	\begin{thm}\label{teo4.3}
		There are exactly 12 non-isomorphic unicyclic graphs $H$ of order $n\leq7$ containing $C_3$ with $\chi_L(H)=n-3$.
	\end{thm}
	
	\begin{proof}
		Let $H$ be a unicyclic graph of order $n\leq 7$ containing $C_3$. Since the order of $H$ must be at least $6$, then $H$ must be a connected graph obtained from three rooted trees of total order $n=6$ or $n=7$, by connecting all roots into such a cycle $C_3$. By Corollary \ref{cor2.2}, the diameter of $H$ is at least $2$ and at most $5$. These restrictions lead to 25 possible graphs $H$ up to isomorphism, as shown in Figure 7 with their minimum locating-colorings. 
		Thus, there are only 12 of them having the required locating-chromatic number (inside the blue square). 
	\end{proof}
	
	\begin{figure}[!ht]\label{fig7}
	\begin{center}
		\includegraphics[scale=.116]{Fig7.png} %0.156
		\caption{All possible graphs $H$ of order $n\leq7$ containing $C_3$ with their minimum locating-colorings.}
	\end{center}
	\end{figure}

	\begin{thm}\label{teo4.4}
		There are exactly 8 non-isomorphic unicyclic graphs $H$ of order $n \leq 7$ containing $C_4$ with $\chi_L(H)=n-3$.
	\end{thm}
	
	\begin{proof}
		Let $H$ be a unicyclic graph of order $n\leq 7$ containing $C_4$. Since the order of $H$ must be at least $6$, then $H$ must be a connected graph obtained from three rooted trees of total order $n=6$ or $n=7$, by connecting all roots into such a cycle $C_4$. By Corollary \ref{cor2.2}, the diameter of $H$ is at least $2$ and at most $5$. These restrictions lead to 13 possible graphs $H$ up to isomorphism, as shown in Figure 8 with their minimum locating-colorings.
		Thus, there are only 8 of them having the required locating-chromatic number (inside the
		blue square), hence it completes the proof.
	\end{proof}

	\begin{figure}[!ht]\label{fig8}
		\begin{center}
			\includegraphics[scale=.15]{Fig8.png} %0.2
			\caption{All possible graphs $H$ of order $n\leq7$ containing $C_4$ with their minimum locating-colorings.}
		\end{center}
	\end{figure}


\section*{Acknowledgement}
This research has been supported by Research Grant “{\em Program Penelitian dan Pengabdian kepada Masyarakat (P3MI)}", Institut Teknologi Bandung, Indonesia.

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%\section*{References}	
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	\bibitem{Asmi-12} Asmiati and E.T. Baskoro, Characterizing all graphs containing cycles with locating-chromatic number 3, {\em AIP Conf. Proc.} {\bf 1450} (2012), 351--357
	
	\bibitem{ASB2019} H. Assiyatun, D.K. Syofyan, E.T. Baskoro, Calculating an upper bound of the locating-chromatic number of trees, {\em Theoretical Computer Science} {\bf 806} (2020), 305--309 
		
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	\bibitem{Dudenko-23-17} M. Dudenko and O. Bogdana, On unicyclic graph of metric dimension 2, \textit{Algebra and Discrete Math.} \textbf{23} (2017), 216--222
	
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\end{thebibliography}

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