arXiv subjects

Sebastián Bustamante

Publications and source records attributed to Sebastián Bustamante.

7 recordsLinked to original sources

arXiv2022

Clique immersions and independence number

The analogue of Hadwiger's conjecture for the immersion order states that every graph $G$ contains $K_{χ(G)}$ as an immersion. If true, it would imply that every graph with $n$ vertices and independence number $α$ contains $K_{\lceil \frac nα\rceil}$ as an immersion. The best currently known bound for this conjecture is due to Gauthier, Le and Wollan, who recently proved that every graph $G$ contains an immersion of a clique on $\bigl\lceil \frac{χ(G)-4}{3.54}\bigr\rceil$ vertices. Their result implies that every $n$-vertex graph with independence number $α$ contains an immersion of a clique on $\bigl\lceil \frac{n}{3.54α}-1.13\bigr\rceil$ vertices. We improve on this result for all $α\ge 3$, by showing that every $n$-vertex graph with independence number $α\ge 3$ contains an immersion of a clique on $\bigl\lfloor \frac {n}{2.25 α- f(α)} \bigr\rfloor - 1$ vertices, where $f$ is a nonnegative function.

math.CO↗

arXiv2020

Partitioning edge-coloured hypergraphs into few monochromatic tight cycles

Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-coloured complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for for all natural numbers $p$ and $r$, the vertices of every $r$-edge-coloured complete graph can be partitioned into a bounded number of $p$-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further extends these results to all host hypergraphs of bounded independence number.

math.CO↗

arXiv2020

Galaxy pairs in the Sloan Digital Sky Survey -- XIV. Galaxy mergers do not lie on the Fundamental Metallicity Relation

In recent observational studies, star-forming galaxies have been shown to follow a relation often dubbed the fundamental metallicity relation (FMR). This relation links the stellar mass of a galaxy with its star formation rate (SFR) and its gas-phase metallicity. Specifically, the FMR predicts that galaxies, at a given stellar mass, exhibit lower metallicities for higher SFRs. This trend is qualitatively consistent with observations of galaxy pairs, which have been robustly shown to experience increasing gas-phase metallicity dilution and enhanced star formation activity with decreasing projected separation. In this work, we show that, despite the qualitative consistency with FMR expectations, the observed O/H dilution in galaxy pairs of the Sloan Digital Sky Survey is stronger than what is predicted by the FMR. We conclude that the evolutionary phase of galaxies interacting with companions is not encoded in the FMR, and thus, mergers constitute a clearly defined population of outliers. We find that galaxies in pairs are consistent with the FMR only when their separation is larger than 110 kpc. Finally, we also quantify the local environment of the pairs using the number of galaxy neighbours within $2\, \mathrm{Mpc}$, $N_2$, and the projected separation to the second closest galaxy, $r_2$. We find that pairs are more sensitive to a second companion than to the local galaxy density, displaying less elevated SFRs with smaller values of $r_2$.

astro-ph.GA↗

arXiv2019

Partitioning infinite hypergraphs into few monochromatic Berge-paths

Extending a result of Rado to hypergraphs, we prove that for all $s, k, t \in \mathbb{N}$ with $k \geq t \geq 2$, the vertices of every $r = s(k-t+1)$-edge-coloured countably infinite complete $k$-graph can be partitioned into the cores of at most $s$ monochromatic $t$-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible.

math.CO↗

arXiv2019

The Auriga Stellar Haloes: Connecting stellar population properties with accretion and merging history

We examine the stellar haloes of the Auriga simulations, a suite of thirty cosmological magneto-hydrodynamical high-resolution simulations of Milky Way-mass galaxies performed with the moving-mesh code AREPO. We study halo global properties and radial profiles out to $\sim 150$ kpc for each individual galaxy. The Auriga haloes are diverse in their masses and density profiles; mean metallicity and metallicity gradients; ages; and shapes, reflecting the stochasticity inherent in their accretion and merger histories. A comparison with observations of nearby late-type galaxies shows very good agreement between most observed and simulated halo properties. However, Auriga haloes are typically too massive. We find a connection between population gradients and mass assembly history: galaxies with few significant progenitors have more massive haloes, possess large negative halo metallicity gradients and steeper density profiles. The number of accreted galaxies, either disrupted or under disruption, that contribute 90% of the accreted halo mass ranges from 1 to 14, with a median of 6.5, and their stellar masses span over three orders of magnitude. The observed halo mass--metallicity relation is well reproduced by Auriga and is set by the stellar mass and metallicity of the dominant satellite contributors. This relationship is found not only for the accreted component but also for the total (accreted + in-situ) stellar halo. Our results highlight the potential of observable halo properties to infer the assembly history of galaxies.

astro-ph.GA↗

arXiv2018

Monochromatic tree covers and Ramsey numbers for set-coloured graphs

We consider a generalisation of the classical Ramsey theory setting to a setting where each of the edges of the underlying host graph is coloured with a {\em set} of colours (instead of just one colour). We give bounds for monochromatic tree covers in this setting, both for an underlying complete graph, and an underlying complete bipartite graph. We also discuss a generalisation of Ramsey numbers to our setting and propose some other new directions. Our results for tree covers in complete graphs imply that a stronger version of Ryser's conjecture holds for $k$-intersecting $r$-partite $r$-uniform hypergraphs: they have a transversal of size at most $r-k$. (Similar results have been obtained by Király et al., see below.) However, we also show that the bound $r-k$ is not best possible in general.

math.CO↗

arXiv2017

Origin of chemically distinct discs in the Auriga cosmological simulations

The stellar disk of the Milky Way shows complex spatial and abundance structure that is central to understanding the key physical mechanisms responsible for shaping our Galaxy. In this study, we use six very high resolution cosmological zoom simulations of Milky Way-sized haloes to study the prevalence and formation of chemically distinct disc components. We find that our simulations develop a clearly bimodal distribution in the $[\rm α/Fe]$ -- $[\rm Fe/H]$ plane. We find two main pathways to creating this dichotomy which operate in different regions of the galaxies: a) an early ($z>1$) and intense high-$\rm[α/Fe]$ star formation phase in the inner region ($R\lesssim 5$ kpc) induced by gas-rich mergers, followed by more quiescent low-$\rm[α/Fe]$ star formation; and b) an early phase of high-$\rm[α/Fe]$ star formation in the outer disc followed by a shrinking of the gas disc owing to a temporarily lowered gas accretion rate, after which disc growth resumes. In process b), a double-peaked star formation history around the time and radius of disc shrinking accentuates the dichotomy. If the early star formation phase is prolonged (rather than short and intense), chemical evolution proceeds as per process a) in the inner region, but the dichotomy is less clear. In the outer region, the dichotomy is only evident if the first intense phase of star formation covers a large enough radial range before disc shrinking occurs; otherwise, the outer disc consists of only low-$\rm[α/Fe]$ sequence stars. We discuss the implication that both processes occurred in the Milky Way.

astro-ph.GA↗