Searcharxiv⌕ Search

arXiv subjects

Rohan Kapadia

Publications and source records attributed to Rohan Kapadia.

9 recordsLinked to original sources
arXiv2020

Densities of minor-closed graph classes are rational

For a graph class $\mathcal{F}$, let $ex_{\mathcal{F}}(n)$ denote the maximum number of edges in a graph in $\mathcal{F}$ on $n$ vertices. We show that for every proper minor-closed graph class $\mathcal{F}$ the function $ex_{\mathcal{F}}(n) - \Delta n$ is eventually periodic, where $\Delta = \lim_{n \to \infty} ex_{\mathcal{F}}(n)/n$ is the limiting density of $\mathcal{F}$. This confirms a special case of a conjecture by Geelen, Gerards and Whittle. In particular, the limiting density of every proper minor-closed graph class is rational, which answers a question of Eppstein. As a major step in the proof we show that every proper minor-closed graph class contains a subclass of bounded pathwidth with the same limiting density, confirming a conjecture of the second author. Finally, we investigate the set of limiting densities of classes of graphs closed under taking topological minors.

math.CO↗
arXiv2019

Asymptotic Density of Graphs Excluding Disconnected Minors

For a graph $H$, let $$c_{\infty}(H)= \lim_{n \to \infty}\max\frac{|E(G)|}{n},$$ where the maximum is taken over all graphs $G$ on $n$ vertices not containing $H$ as a minor. Thus $c_{\infty}(H)$ is the asymptotic maximum density of graphs not containing $H$ as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on $c_{\infty}(H)$ for disconnected graphs $H$. In particular, we determine $c_{\infty}(H)$ whenever $H$ is union of cycles. Finally, we investigate the behaviour of $c_\infty(sK_r)$ for fixed $r$, where $sK_r$ denotes the union of $s$ disjoint copies of the complete graph on $r$ vertices. Improving on a result of Thomason, we show that $$c_\infty(sK_r)=s(r-1)-1 \mathrm{\; for \;} s =Ω\left(\frac{\log{r}}{\log\log{r}}\right),$$ and $$c_\infty(sK_r)>s(r-1)-1 \mathrm{\; for \;} s ={o}\left(\frac{\log{r}}{\log\log{r}}\right).$$

math.CO↗
arXiv2016

The extremal functions of classes of matroids of bounded branch-width

For a set of matroids $\mathcal{M}$, let $ex_\mathcal{M}(n)$ be the maximum size of a simple rank-$n$ matroid in $\mathcal{M}$. We prove that, for any finite field $\mathbb{F}$, if $\mathcal{M}$ is a minor-closed class of $\mathbb{F}$-representable matroids of bounded branch-width, then $\lim_{n \rightarrow \infty} ex_\mathcal{M}(n) / n$ exists and is a rational number, $Δ$. We also show that $ex_\mathcal{M}(n) - Δn$ is periodic when $n$ is sufficiently large and that $ex_\mathcal{M}$ is achieved by a subclass of $\mathcal{M}$ of bounded path-width.

math.CO↗
arXiv2015

Representability of matroids with a large projective geometry minor

We prove that for each prime power $q$ there is an integer $n$ such that if $M$ is a $3$-connected, representable matroid with a PG$(n-1,q)$-minor and no $U_{2,q^2+1}$-minor, then $M$ is representable over GF$(q)$. We also show that for $\ell >= 2$, if $M$ is a $3$-connected, representable matroid of sufficiently high rank with no $U_{2,\ell+2}$-minor and $|E(M)| \geq (4\ell)^{r(M)/2}$, then $M$ is representable over a field of order at most $\ell$.

math.CO↗
arXiv2014

Lines, betweenness and metric spaces

A classic theorem of Euclidean geometry asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with a certain natural definition of lines in a metric space. We prove that in any metric space with $n$ points, either there is a line containing all the points or there are at least $Ω(\sqrt{n})$ lines. This is the first polynomial lower bound on the number of lines in general finite metric spaces. In the more general setting of pseudometric betweenness, we prove a corresponding bound of $Ω(n^{2/5})$ lines. When the metric space is induced by a connected graph, we prove that either there is a line containing all the points or there are $Ω(n^{4/7})$ lines, improving the previous $Ω(n^{2/7})$ bound. We also prove that the number of lines in an $n$-point metric space is at least $n / 5w$, where $w$ is the number of different distances in the space, and we give an $Ω(n^{4/3})$ lower bound on the number of lines in metric spaces induced by graphs with constant diameter, as well as spaces where all the positive distances are from \{1, 2, 3\}.

math.CO↗
arXiv2014

The Chen-Chvátal conjecture for metric spaces induced by distance-hereditary graphs

A special case of a theorem of De Bruijn and Erdős asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvátal conjectured a generalization of this result to arbitrary finite metric spaces, with a particular definition of lines in a metric space. We prove it for metric spaces induced by connected distance-hereditary graphs -- a graph $G$ is called distance-hereditary if the distance between two vertices $u$ and $v$ in any connected induced subgraph $H$ of $G$ is equal to the distance between $u$ and $v$ in $G$.

math.MG↗
arXiv2014

Matroids with a modular 4-point line

A result of Seymour implies that any 3-connected matroid with a modular 3-point line is binary. We prove a similar characterization for 3-connected matroids with modular 4-point lines. We show that such a matroid is either representable over GF(3) or GF(4) or has an $F_7$-minor and either an $(F_7^-)$- or $(F_7^-)^*$-minor.

math.CO↗
arXiv2013

Representation of matroids with a modular plane

We prove that if M is a vertically 4-connected matroid with a modular flat X of rank at least three, then every representation of M | X over a finite field F extends to a unique F-representation of M. A corollary is that when F has order q, any vertically 4-connected matroid with a PG(2, F)-restriction is either F-representable or has a U_{2, q^2+1}-minor. We also show that no excluded minor for the class of F-representable matroids has a PG(2, F)-restriction.

math.CO↗