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%% define your title in the usual way
\title[Faces of Directed Edge Polytopes]{Faces of Directed Edge Polytopes}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
% \author[Optional shorter names]{Longer names me\thanks{\href{mailto:hello@world.c}{hello@world.c}. Longer names me was partially supported by Grant 2017.11.14.$\partial$\;supp.}\addressmark{1}, \and you\addressmark{2}}
\author[Y. Numata, Y. Takahashi, D. Tamaki]{%
  Yasuhide NUMATA\thanks{%
    \href{mailto:nu@math.sci.hokudai.ac.jp}{nu@math.sci.hokudai.ac.jp}.
    The first author was partially supported by JSPS KAKENHI Grant Number JP18K03206.}\addressmark{1},
  \and
  Yusuke TAKAHASHI\addressmark{2}
  \and
  Dai Tamaki\thanks{%
    \href{mailto:rivulus@shinshu-u.ac.jp}{rivulus@shinshu-u.ac.jp}.
    The third author was partially supported by JSPS KAKENHI Grant
    Number JP20K03579.
  }\addressmark{3}
}

%% then use \addressmark to match authors to institutions here
%\address{\addressmark{1}Department of This, University of That, There \\ \addressmark{2}School of Something, University of Something Else, Somewhere}
\address{%
  \addressmark{1}Department of Mathematics, Hokkaido University, Sapporo, Japan,\\
  \addressmark{2}Graduate School of Science and Technology, Shinshu University, Matsumoto, Japan,\\
  \addressmark{3}Department of Mathematics, Shinshu University, Matsumoto, Japan.}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{ Given a finite quiver (directed graph) without loops and
 multiedges, the convex hull of the column vector of the incidence
 matrix is called the directed edge polytope and is an interesting
 example of lattice polytopes. 
 In this article, we give a complete characterization of facets of the
 directed edge polytope of an arbitrary finite quiver without loops and
 multiedges in terms of the connectivity and the existence of a rank
 function. Our result can be regarded as an extension of the result of
 Higashitani et al.~\cite{MR3949939} on facets of symmetric edge
 polytopes to directed edge polytopes.
 When the quiver in question has a rank function, we obtain a
 characterization of faces of arbitrary dimensions.
 This article is an exteded abstract of \cite{2203.14521}.
}

%% put your French abstract here, or comment this out if you don't have one
\resume{
Pour un graphe orienté fini simple,
l'enveloppe convexe du vecteur colonne de la matrice d'incidence est appelée le polytope d'arête orienté.
C'est un intéressant
exemple de polytopes intégraux.
Dans cet article, nous donnons une caractérisation complète des facettes
de polytope d'arête orienté
en termes de connectivité et d'existence d'une fonction de rang.
Notre résultat peut être considéré comme une extension du résultat
de Higashitani et al.~\cite{MR3949939}
sur les facettes de polytope d'arête symétrique.
Pour un graphe orienté avec une fonction de rang,
on obtient une caractérisation de faces de dimensions arbitraires.
Cet article est un résumé étendu de \cite{2203.14521}.
}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Kantorovich--Rubinstein polytopes; fundamental polytopes; symmetric edge polytopes; directed edge polytopes; characterization of faces; $f$-vectors}
%\subjclass[2020]{52B05}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex]{biblatex}
%\addbibresource{sample.bib}
\addbibresource{by-mr.bib}
\addbibresource{by-arxiv.bib}

%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}



\maketitle


% Introduction
\input{p1.tex}
% Main part
%\input{p2.tex}
\input{p21.tex}
\input{p22.tex}
\input{p23.tex}
\input{p24.tex}

% References
\printbibliography
\end{document}

