Office 412, Edifici Omega, Campus Nord

Carrer Jordi Girona 1-3

08034 Barcelona

jordi.castellvi@upc.edu

Hello! Welcome to my personal webpage.

I'm 26 years old and I live in Cornellà de Llobregat. I'm interested in mathematics and politics.

I'm a PhD student in the GAPCOMB research group of the Mathematics Department in Universitat Politècnica de Catalunya (UPC), under the supervision of Marc Noy and Clément Requilé.

I've mainly worked on enumerative combinatorics of graphs and maps, both from a bijective and an analytic viewpoint.

Chek out my Curriculum Vitae.

**Bijective enumeration of constellations in higher genus** (febrer del 2021).

Bachelor's thesis written under the supervision of Marie Albenque and Éric Fusy.

*Universitat Politècnica de Catalunya* i *École Polytechnique*.

Report,
slides.

Published in a short format at *Reports@SCM* (link).

Bousquet-Mélou and Schaeffer gave in 2000 a bijective enumeration of some planar maps called constellations. In 2019, Lepoutre described a bijection between bicolorable maps of arbitrary genus and some unicellular maps of the same genus.

We present a bijection between constellations of higher genus and some unicellular maps that generalizes both existing bijections at the same time.

Using this bijection, we manage to enumerate a subclass of constellations on the torus, proving that its generating function is a rational function of the generating function of some trees.

**Enumeration of chordal planar graphs and maps** (maig del 2022).

Master's thesis written under the supervision of Marc Noy and Clément Requilé.

*Universitat Politècnica de Catalunya*.

Report,
slides.

We determine the number of labelled chordal planar graphs with \(n\) vertices, which is asymptotically \(c_1\cdot n^{-5/2} \gamma^n n!\) for a constant \(c_1>0\) and \(\gamma \approx 11.89235\). We also determine the number of rooted simple chordal planar maps with \(n\) edges, which is asymptotically \(c_2 n^{-3/2} \delta^n\), where \(\delta = 1/\sigma \approx 6.40375\), and \(\sigma\) is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from \(K_4\) by repeatedly adding vertices adjacent to an existing triangular face.

**Limits of chordal graphs with bounded tree-width**,

with Benedikt Stufler.

*Submitted.* arXiv:2310.20423.

We study random \(k\)-connected chordal graphs with bounded tree-width. Our main results are scaling limits and quenched local limits.

**Chordal graphs with bounded tree-width**,

with Michael Drmota, Marc Noy and Clément Requilé.

*Advances in Applied Mathematics, Volume 157, 2024.* arXiv:2301.00194.

Conference version.

Given \(t\geq 2\) and \(0\leq k\leq t\), we prove that the number of labelled \(k\)-connected chordal graphs with \(n\) vertices and tree-width at most \(t\) is asymptotically \(c n^{-5/2} \gamma^n n!\), as \(n\to\infty\), for some constants \(c,\gamma >0\) depending on \(t\) and \(k\). Additionally, we show that the number of \(i\)-cliques (\(2\leq i\leq t\)) in a uniform random \(k\)-connected chordal graph with tree-width at most \(t\) is normally distributed as \(n\to\infty\).

The asymptotic enumeration of graphs of tree-width at most \(t\) is wide open for \(t\geq 3\). To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, *Graphs and Combinatorics* (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on \(n\) vertices.

**Enumeration of chordal planar graphs and maps**,

with Marc Noy and Clément Requilé.

*Discrete Mathematics, Volume 346, Issue 1, 2023.* arXiv:2202.13340.

We determine the number of labelled chordal planar graphs with \(n\) vertices, which is asymptotically \(c_1\cdot n^{-5/2} \gamma^n n!\) for a constant \(c_1>0\) and \(\gamma \approx 11.89235\). We also determine the number of rooted simple chordal planar maps with \(n\) edges, which is asymptotically \(c_2 n^{-3/2} \delta^n\), where \(\delta = 1/\sigma \approx 6.40375\), and \(\sigma\) is an algebraic number of degree 12. The proofs are based on combinatorial decompositions and singularity analysis. Chordal planar graphs (or maps) are a natural example of a subcritical class of graphs in which the class of 3-connected graphs is relatively rich. The 3-connected members are precisely chordal triangulations, those obtained starting from \(K_4\) by repeatedly adding vertices adjacent to an existing triangular face.

**Chordal graphs with bounded tree-width**,

with Michael Drmota, Marc Noy and Clément Requilé.

*Eurocomb 2023.*

PDF,
proceedings,
journal version.

Given $t\geq 2$ and $0\leq k\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} \gamma^n n!$, as $n\to\infty$, for some constants $c,\gamma >0$ depending on $t$ and $k$. Additionally, we show that the number of $i$-cliques ($2\leq i\leq t$) in a uniform random $k$-connected chordal graph with tree-width at most $t$ is normally distributed as $n\to\infty$.\n The asymptotic enumeration of graphs of tree-width at most $t$ is wide open for $t\geq 3$. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, *Graphs and Combinatorics* (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on $n$ vertices.

**Enumeration of unlabelled chordal graphs with bounded tree-width**,

with Clément Requilé.

*DMD 2024.*

PDF.

**Enumeration of chordal planar graphs and maps** (22/03/2022).

Journées ALEA 2022.

*Centre International de Rencontres Mathématiques - Luminy*.

Slides.

A graph is chordal if it has no induced cycle of length greater than three. This talk is mainly devoted to the enumerative study of labelled chordal planar graphs. We follow the canonical decomposition of graphs into $k$-connected components for $k = 1, 2, 3$ and we make use of the dissymmetry theorem to keep our proofs purely combinatorial. The 3-connected graphs in this family are precisely chordal triangulations: the ones obtained starting from a $K_4$ and repeatedly adding vertices adjacent to an existing triangular face. We then consider networks, which are 2-connected labelled chordal planar graphs rooted at a directed edge whose vertices have no labels. From the generating function of networks, we obtain the generating functions of 2-connected, connected and, finally, general labelled chordal planar graphs. The singularity analysis of the resulting equations leads to our main result: the number of labelled chordal planar graphs with $n$ vertices is asymptotically $c_1 n^{-5/2} \gamma^n n!$, where $c_1 > 0$ is a constant and $\gamma \approx 11.89235$. Using the same techniques, we also determine the number of rooted simple chordal planar maps with $n$ edges, which is asymptotically $c_2 n^{-3/2} \delta^n$, where $\delta = 1 / \sigma \approx 6.40375$, and $\sigma$ is an algebraic number of degree 12.\n Work in collaboration with Marc Noy and Clément Requilé.

**Chordal graphs with bounded tree-width** (31/08/2023).

Eurocomb 2023.

*Charles University - Prague*.

Slides.

Given $t\geq 2$ and $0\leq k\leq t$, we prove that the number of labelled $k$-connected chordal graphs with $n$ vertices and tree-width at most $t$ is asymptotically $c n^{-5/2} \gamma^n n!$, as $n\to\infty$, for some constants $c,\gamma >0$ depending on $t$ and $k$. Additionally, we show that the number of $i$-cliques ($2\leq i\leq t$) in a uniform random $k$-connected chordal graph with tree-width at most $t$ is normally distributed as $n\to\infty$.\n The asymptotic enumeration of graphs of tree-width at most $t$ is wide open for $t\geq 3$. To the best of our knowledge, this is the first non-trivial class of graphs with bounded tree-width where the asymptotic counting problem is solved. Our starting point is the work of Wormald [Counting Labelled Chordal Graphs, *Graphs and Combinatorics* (1985)], were an algorithm is developed to obtain the exact number of labelled chordal graphs on $n$ vertices.

**Enumeration of unlabelled chordal graphs with bounded tree-width** (04/07/2024).

DMD 2024.

*Universidad de Alcalá - Alcalá de Henares*.

Slides.

You can write me at jordi.castellvi@upc.edu for professional matters and at jordi@jcastellvi.net for the rest.

Cheers!

Last update: 05/07/2024.