\documentclass[12pt]{article} \usepackage{e-jc} \specs{P1.70}{22(1)}{2015} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} %\newtheorem{lemma}{Lemma} %\newtheorem{theorem}{Theorem} \newcommand{\dist}{\mbox{\rm dist\/}} %\setlength{\parindent}{0pt} %\setlength{\parskip}{2ex} \title{\bf A De Bruijn--Erd\H os theorem for chordal graphs} \author{ Laurent Beaudou\\ \small Universit\' e Blaise Pascal\\[-0.8ex] \small Clermont-Ferrand, France\\ \small \texttt{laurent.beaudou@univ-bpclermont.fr}\\ \and Adrian Bondy\\ \small Universit\' e Paris 6\\[-0.8ex] \small Paris, France\\ \small \texttt{adrian.bondy@sfr.fr}\\ \and Xiaomin Chen\\ \small Shanghai Jianshi Ltd\\[-0.8ex] \small Shanghai, P.R.~China\\ \small \texttt{gougle@gmail.com}\\ \and Ehsan Chiniforooshan\\ \small Google Kitchener-Waterloo\\[-0.8ex] \small Waterloo, Canada\\[-0.8ex] \small \texttt{chiniforooshan@alumni.uwaterloo.ca}\\ \and Maria Chudnovsky\footnote{Partially supported by NSF grants DMS-1001091 and IIS-1117631.}\\ \small Columbia University\\[-0.8ex] \small New York, U.S.A.\\ \small \texttt{mchudnov@columbia.edu}\\ \and Va\v sek Chv\' atal\footnote{Canada Research Chair in Discrete Mathematics.}\\ \small Concordia University\\[-0.8ex] \small Montreal, Canada\\ \small \texttt{chvatal@cse.concordia.ca}\\ \and \phantom{1cm}Nicolas Fraiman\phantom{2cm}\\ \small McGill University\\[-0.8ex] \small Montreal, Canada\\ \small \texttt{nfraiman@gmail.com}\\ \and Yori Zwols\\ \small Concordia University\\[-0.8ex] \small Montreal, Canada\\ \small \texttt{yzwols@gmail.com} } \date{\dateline{Jul 1, 2013}{Mar 11, 2015}{Mar 23, 2015}\\ \small Mathematics Subject Classifications: 05C12, 52C10, 05D05} \begin{document} \maketitle \begin{abstract} A special case of a combinatorial theorem of De Bruijn and Erd\H os asserts that every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chv\'{a}tal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs. \bigskip \noindent \textbf{Keywords:} Combinatorial geometry; Metric space; Extremal combinatorics \end{abstract} \section{Introduction}\label{sec.intro} It is well known that \begin{itemize} \item[(i)] {\em every noncollinear set of $n$ points in the plane determines at least $n$ distinct lines.\/} \end{itemize} As noted by Erd\H{o}s~\cite{E43}, theorem (i) is a corollary of the Sylvester--Gallai theorem (asserting that, for every noncollinear set $S$ of finitely many points in the plane, some line goes through precisely two points of $S$); it is also a special case of a combinatorial theorem proved later by De Bruijn and Erd\H os \cite{DE48}.\\ Theorem (i) involves neither measurement of distances nor measurement of angles: the only notion employed here is incidence of points and lines. Such theorems are a part of {\em ordered geometry\/} \cite{C61}, which is built around the ternary relation of {\em betweenness\/}: point $b$ is said to lie between points $a$ and $c$ if $b$ is an interior point of the line segment with endpoints $a$ and $c$. It is customary to write $[abc]$ for the statement that $b$ lies between $a$ and $c$. In this notation, a {\em line\/} $\overline{uv}$ is defined --- for any two distinct points $u$ and $v$ --- as \begin{equation}\label{def.first} \{u,v\}\;\cup\; \{p: [puv] \vee [upv]\vee [uvp]\}. \end{equation} In terms of the Euclidean metric $\dist$, we have %\begin{multline}\label{def.btw} %\mbox{ $[abc] \;\Leftrightarrow\;$}\\ %\mbox{ $a,b,c$ are three distinct points and %$\dist(a,b)+\dist(b,c) = \dist(a,c)$.} %\end{multline} \begin{equation}\label{def.btw} [abc] \; \Leftrightarrow\; a,b,c \text{ are three distinct points and } \dist(a,b)+\dist(b,c) = \dist(a,c). \end{equation} In an arbitrary metric space, equivalence (\ref{def.btw}) defines the ternary relation of {\em metric betweenness} introduced in \cite{Men} and further studied in \cite{Blu,Bus,Chv04}; in turn, (\ref{def.first}) defines the line $\overline{uv}$ for any two distinct points $u$ and $v$ in the metric space. The resulting family of lines may have strange properties. For instance, a line can be a proper subset of another: in the metric space with points $u,v,x,y,z$ and \begin{align*} &\dist(u,v)=\dist(v,x)=\dist(x,y)=\dist(y,z)=\dist(z,u)=1,\\ &\dist(u,x)=\dist(v,y)=\dist(x,z)=\dist(y,u)=\dist(z,v)=2, \end{align*} we have \[ \overline{vy}=\{v,x,y\} \;\;\mbox{ and }\;\; \overline{xy}=\{v,x,y,z\}. \] \bigskip Chen \cite{Che} proved, using a definition of $\overline{uv}$ different from~(\ref{def.first}), that the Sylvester--Gallai theorem generalizes in the framework of metric spaces. Chen and Chv\'{a}tal \cite{CC08} suggested that theorem (i), too, might generalize in this framework: \begin{itemize} \item[(ii)] {\em True or false? Every metric space on $n$ points, where $n\ge 2$, either has at least $n$ distinct lines or else has a line that consists of all $n$ points.} \end{itemize} They proved that \begin{itemize} \item every metric space on $n$ points either has at least $\lg n$ distinct lines or else has a line that consists of all $n$ points \end{itemize} and noted that the lower bound $\lg n$ can be improved to $\lg n + \frac{1}{2}\lg\lg n + \frac{1}{2}\lg\frac{\pi}{2}- o(1)$. (Here, as usual, $\lg x$ stands for $\log_2 x$.)\\ Every connected undirected graph induces a metric space on its vertex set, where $\dist(u,v)$ is the familiar graph-theoretic distance between vertices $u$ and $v$, defined as the smallest number of edges in a path from $u$ to $v$. (Some people call this the `hop distance'.) Chiniforooshan and Chv\'{a}tal \cite{CC11} proved that \begin{itemize} \item every metric space induced by a connected graph on $n$ vertices either has $\Omega(n^{2/7})$ distinct lines or else has a line that consists of all $n$ vertices; \end{itemize} we will prove that the answer to (ii) is `true' for all metric spaces induced by connected chordal graphs. (We follow the graph-theoretic terminology of Bondy and Murty~\cite{BM}. In particular, a {\em chordal graph\/} is a graph that contains no induced cycle of length four or more.) \begin{theorem}\label{main} Every metric space induced by a connected chordal graph on $n$ vertices, where $n\ge 2$, either has at least $n$ distinct lines or else has a line that consists of all $n$ vertices. \end{theorem} \section{The proof}\label{sec.proof} Given an undirected graph, let us write $[abc]$ to mean that $a,b,c$ are three distinct vertices such that $\dist(a,b)+\dist(b,c)=\dist(a,c)$; this is equivalent to saying that $b$ is an interior vertex of a shortest path from $a$ to $c$. \begin{lemma}\label{lem1} Let $s,x,y$ be vertices in a finite chordal graph such that $[sxy]$. If $\overline{sx}= \overline{sy}$, then $x$ is a cut vertex separating $s$ and $y$. \end{lemma} \begin{proof} The set of all vertices $u$ such that $\dist(s,u)=\dist(s,x)$ separates $s$ and $y$. Among all its subsets that separate $s$ and $y$, choose a minimal one and call it $C$. Since $x$ is an interior vertex of a shortest path from $s$ to $y$, it belongs to $C$. To prove that $C$ includes no other vertex, assume, to the contrary, that $C$ includes a vertex $u$ other than $x$.\\ Our graph with $C$ removed has distinct connected components $S$ and $Y$ such that $s\in S$ and $y\in Y$; the minimality of $C$ guarantees that each of its vertices has at least one neighbour in $S$ and at least one neighbour in $Y$. Since each of $u$ and $x$ has at least one neighbour in $S$, there is a path from $u$ to $x$ with at least one interior vertex and with all interior vertices in $S$. Let $P$ be a shortest such path; note that $P$ has no chords except possibly the chord $ux$. Similarly, there is a path $Q$ from $u$ to $x$ with at least one interior vertex, and with all interior vertices in $Y$, that has no chords except possibly the chord $ux$. The union of $P$ and $Q$ is a cycle of length at least four; since this cycle must have a chord, vertices $u$ and $x$ must be adjacent. In turn, the union of $Q$ and $ux$ is a chordless cycle, and so $Q$ has precisely two edges. This means that some vertex $v$ in $Y$ is adjacent to both $u$ and $x$. (Similarly, some vertex in $S$ is adjacent to both $u$ and $x$; however, this fact is irrelevant to our argument.)\\ Write $i=\dist(s,x)$ and $j=\dist(x,y)$. Since all vertices $t$ with $\dist(s,t)i$; since $\dist(x,v)=1$, we conclude that $\dist(s,v)=i+1$ and that $v\in \overline{sx}$. Since $\overline{sx}=\overline{sy}$, it follows that $v\in \overline{sy}$. Since $\dist(v,x)=1$ and $\dist(x,y)=j$, we have $\dist(v,y)\le j+1$. From $\dist(s,v)=i+1$, $\dist(s,y)=i+j$, $\dist(v,y)\le j+1$, $i\ge 1$, $j\ge 1$, and $v\in \overline{sy}$, we deduce that $\dist(v,y)=j-1$.\\ Since $\dist(u,v)=1$, it follows that $\dist(u,y)\le j$; since $\dist(s,u)=i$ and $\dist(s,y)=i+j$, we conclude that $\dist(u,y)=j$ and $u\in \overline{sy}$. Since $\dist(s,u)=i$, $\dist(s,x)=i$, and $\dist(u,x)=1$, we have $u\not\in \overline{sx}$. But then $\overline{sx}\ne \overline{sy}$, a contradiction. \end{proof} A vertex of a graph is called {\em simplicial\/} if its neighbours are pairwise adjacent. \begin{lemma}\label{lem2} Let $s,x,y$ be three distinct vertices in a finite connected chordal graph. If $s$ is simplicial and $\overline{sx}= \overline{sy}$, then $\overline{xy}$ consists of all the vertices of the graph. \end{lemma} \begin{proof} Since $\overline{sx}= \overline{sy}$, we have $y\in\overline{sx}$, and so $[ysx]$ or $[syx]$ or $[sxy]$; since $s$ is simplicial, $[ysx]$ is excluded; switching $x$ and $y$ if necessary, we may assume that $[sxy]$. Given an arbitrary vertex $u$, we have to prove that $u\in\overline{xy}$. Let $P$ be a shortest path from $s$ to $u$ and let $Q$ be a shortest path from $u$ to $y$. Lemma~\ref{lem1} guarantees that $x$ is a cut vertex separating $s$ and $y$, and so the concatenation of $P$ and $Q$ must pass through $x$. This means that $[sxu]$ or $[uxy]$ (or both). If $[uxy]$, then $u\in\overline{xy}$; to complete the proof, we may assume that $[sxu]$, and so $u\in\overline{sx}$.\\ Since $\overline{sx}=\overline{sy}$, we have $[usy]$ or $[suy]$ or $[syu]$; since $s$ is simplicial, $[usy]$ is excluded. If $[suy]$, then $[sxu]$ implies $[xuy]$; if $[syu]$, then $[sxy]$ implies $[xyu]$; in either case, $u\in\overline{xy}$. \end{proof} \begin{proof}[Proof of Theorem~\ref{main}]\ Consider a connected chordal graph on $n$ vertices where $n\ge 2$. By a theorem of Dirac~\cite[Theorem 4]{D}, this graph has at least two simplicial vertices; choose one of them and call it $s$. We may assume that the lines $\overline{sz}$ with $z\ne s$ are pairwise distinct (else some line consists of all $n$ vertices by Lemma~\ref{lem2}). Since the graph is connected and has at least two vertices, $s$ has at least one neighbour; choose one and call it $u$. If $u$ is the only neighbour of $s$, then every path from $s$ to another vertex must pass through $u$, and so $\overline{su}$ consists of all $n$ vertices. If $s$ has a neighbour $v$ other than $u$, then line $\overline{uv}$ is distinct from all of the $n-1$ lines $\overline{sz}$ with $z\ne s$: since $s,u,v$ are pairwise adjacent, we have $s\not\in\overline{uv}$. \end{proof} \section{Related theorems}\label{sec.conc} In Theorem~\ref{main}, `connected chordal graph' can be replaced by `connected bipartite graph': \begin{itemize} \item every metric space induced by a connected bipartite graph on $n$ vertices, where $n\ge 2$, has a line that consists of all $n$ vertices. \end{itemize} In fact, $\overline{xy}$ consists of all $n$ vertices whenever $x$ and $y$ are adjacent. To prove this, consider an arbitrary vertex $u$. Since the graph is bipartite, $\dist(u,x)$ and $\dist(u,y)$ have distinct parities; since $\dist(x,y)=1$, they differ by at most one. We conclude that $\dist(u,x)$ and $\dist(u,y)$ differ by precisely one, and so $u\in\overline{xy}$. \medskip In Theorem~\ref{main}, `connected chordal graph' can be also replaced by `graph of diameter two': Chv\'{a}tal \cite{Chv12} proved that \begin{itemize} \item every metric space on $n$ points where $n\ge 2$ and each nonzero distance equals $1$ or $2$ either has at least $n$ distinct lines or else has a line that consists of all $n$ vertices. \end{itemize} Kantor and Patk\' os~\cite{KP} proved that \begin{itemize} \item if no two of $n$ points in the plane share their $x$- or $y$-coordinate, then these $n$ points with the $L_1$ metric either induce at least $n$ distinct lines or else they induce a line that consists of all of them. \end{itemize} (For sets of $n$ points in the plane that are allowed to share their coordinates, \cite{KP} provides a weaker conclusion: these $n$ points with the $L_1$ metric either induce at least $n/37$ distinct lines or else they induce a line that consists of all of them.) \bigskip \subsection*{Acknowledgment} The work whose results are reported here began at a workshop held at Concordia University in June 2011. 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