arXiv subjects

Silvia Fernández-Merchant

Publications and source records attributed to Silvia Fernández-Merchant.

14 recordsLinked to original sources

arXiv2026

The balanced upper chromatic number of linear hypergraphs and the $n$-cube over $t$ elements

A coloring of the vertices of a hypergraph is called \emph{balanced} if the sizes of the color classes differ by at most one. We say that a hyperedge is \emph{rainbow} if its elements have pairwise distinct colors. In this paper, we provide a general upper bound on the \emph{balanced upper chromatic number} of arbitrary linear hypergraphs, that is, the largest integer $k$ such that there exists a balanced $k$-coloring of the vertices of the hypergraph without rainbow hyperedges. We focus on the cube $C_t^n$, defined as the linear hypergraph whose vertices are the lattice points in $[0,t-1]^n$, and whose hyperedges are the sets of $t$ collinear points. We determine the exact balanced upper chromatic number of $C_t^n$ for $t\geq 4n-2$. For smaller values of $t$, we present bounds and determine this parameter (with few exceptions) in dimensions $2$ and $3$.

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arXiv2024

The rectilinear local crossing number of $K_n$

We determine ${\bar{\rm{lcr}}}(K_n)$, the rectilinear local crossing number of the complete graph $K_n$ for every $n$. More precisely, for every $n \notin \{8, 14 \}, $ \[ {\bar{\rm{lcr}}}(K_n)=\left\lceil \frac{1}{2} \left( n-3-\left\lceil \frac{n-3}{3} \right\rceil \right) \left\lceil \frac{n-3}{3} \right\rceil \right\rceil, \] ${\bar{\rm{lcr}}}(K_8)=4$, and ${\bar{\rm{lcr}}}(K_{14})=15$.

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arXiv2024

Book crossing numbers of the complete graph and small local convex crossing numbers

A $ k $-page book drawing of a graph $ G $ is a drawing of $ G $ on $ k $ halfplanes with common boundary $ l $, a line, where the vertices are on $ l $ and the edges cannot cross $ l $. The $ k $-page book crossing number of the graph $ G $, denoted by $ ν_k(G) $, is the minimum number of edge-crossings over all $ k $-page book drawings of $ G $. Let $G=K_n$ be the complete graph on $n$ vertices. We improve the lower bounds on $ ν_k(K_n) $ for all $ k\geq 14 $ and determine $ ν_k(K_n) $ whenever $ 2 < n/k \leq 3 $. Our proofs rely on bounding the number of edges in convex graphs with small local crossing numbers. In particular, we determine the maximum number of edges that a convex graph with local crossing number at most $ \ell $ can have for $ \ell\leq 4 $.

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arXiv2023

Bounding the tripartite-circle crossing number of complete tripartite graphs

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. We present upper and lower bounds on the minimum number of crossings in tripartite-circle drawings of $K_{m,n,p}$. In contrast to 1- and 2-circle drawings, which may attain the Harary-Hill bound, our results imply that balanced restricted 3-circle drawings of the complete graph are not optimal.

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arXiv2023

The tripartite-circle crossing number of graphs with two small partition classes

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of a tripartite graph is the minimum number of edge crossings among all its tripartite-circle drawings. We determine the exact value of the tripartite-circle crossing number of $K_{a,b,n}$, where $a,b\leq 2$.

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arXiv2022

Direction-Critical Configurations in Noncentral General Position

In 1982, Ungar proved that the connecting lines of a set of $n$ noncollinear points in the plane determine at least $2\lfloor n/2 \rfloor$ directions (slopes). Sets achieving this minimum for $n$ odd (even) are called \emph{direction-(near)-critical} and their full classification is still open. To date, there are four known infinite families and over 100 sporadic critical configurations. Jamison conjectured that any direction-critical configuration with at least 50 points belongs to those four infinite families. Interestingly, except for a handful of sporadic configurations, all these configurations are centrally symmetric. We prove Jamison's conjecture, and extend it to the near-critical case, for centrally symmetric configurations in \emph{noncentral general position}, where only the connecting lines through the center of symmetry may pass through more than two points. As in Ungar's proof, our results are proved in the more general setting of \emph{allowable sequences}. We show that, up to equivalence, the \emph{central signature} of a set uniquely determines a centrally symmetric direction-(near)-critical allowable sequence in noncentral general position, and classify such allowable sequences that are geometrically realizable.

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arXiv2019

$K_{1,3}$-covering red and blue points in the plane

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $R\cup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $R\cup B$, like $\{1,3\}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $R\cup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $R\cup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $b\le r\le 3b$, then at least $\frac{8}{9}(r+b-8)$ points of $R\cup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.

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arXiv2018

Bishellable drawings of $K_n$

The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is $ H(n) = \frac 1 4 \left\lfloor\frac{\mathstrut n}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-1}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-2}{\mathstrut 2}\right\rfloor \left\lfloor\frac{\mathstrut n-3}{\mathstrut 2}\right \rfloor$. Ábrego et al. introduced the notion of shellability of a drawing $D$ of $K_n$. They proved that if $D$ is $s$-shellable for some $s\geq\lfloor\frac{n}{2}\rfloor$, then $D$ has at least $H(n)$ crossings. This is the first combinatorial condition on a drawing that guarantees at least $H(n)$ crossings. In this work, we generalize the concept of $s$-shellability to bishellability, where the former implies the latter in the sense that every $s$-shellable drawing is, for any $b \leq s-2$, also $b$-bishellable. Our main result is that $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellability of a drawing $D$ of $K_n$ also guarantees, with a simpler proof than for $s$-shellability, that $D$ has at least $H(n)$ crossings. We exhibit a drawing of $K_{11}$ that has $H(11)$ crossings, is 3-bishellable, and is not $s$-shellable for any $s\geq5$. This shows that we have properly extended the class of drawings for which the Harary-Hill Conjecture is proved. Moreover, we provide an infinite family of drawings of $K_n$ that are $(\lfloor \frac{n}{2} \rfloor\!-\!2)$-bishellable, but not $s$-shellable for any $s\geq\lfloor\frac{n}{2}\rfloor$.

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arXiv2013

Shellable drawings and the cylindrical crossing number of $K_n$

The Harary-Hill Conjecture States that the number of crossings in any drawing of the complete graph $ K_n $ in the plane is at least $Z(n):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor \frac{n-2}{2}\right\rfloor\left\lfloor \frac{n-3}{2}\right\rfloor$. In this paper, we settle the Harary-Hill conjecture for {\em shellable drawings}. We say that a drawing $D$ of $ K_n $ is {\em $ s $-shellable} if there exist a subset $ S = \{v_1,v_2,\ldots,v_ s\}$ of the vertices and a region $R$ of $D$ with the following property: For all $1 \leq i < j \leq s$, if $D_{ij}$ is the drawing obtained from $D$ by removing $v_1,v_2,\ldots v_{i-1},v_{j+1},\ldots,v_{s}$, then $v_i$ and $v_j$ are on the boundary of the region of $D_{ij}$ that contains $R$. For $ s\geq n/2 $, we prove that the number of crossings of any $ s $-shellable drawing of $ K_n $ is at least the long-conjectured value Z(n). Furthermore, we prove that all cylindrical, $ x $-bounded, monotone, and 2-page drawings of $ K_n $ are $ s $-shellable for some $ s\geq n/2 $ and thus they all have at least $ Z(n) $ crossings. The techniques developed provide a unified proof of the Harary-Hill conjecture for these classes of drawings.

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arXiv2011

On $(\le k)$-edges, crossings, and halving lines of geometric drawings of $K_n$

Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of $P$. A \emph{halving line} of $P$ is a line passing though two points of $P$ that divides the rest of the points of $P$ in (almost) half. The number of halving lines of $P$ is denoted by $h(P)$. Similarly, a $k$\emph{-edge}, $0\leq k\leq n/2-1$, is a line passing through two points of $P$ and leaving exactly $k$ points of $P$ on one side. The number of $(\le k)$-edges of $P$ is denoted by $E_{\leq k}(P) $. Let $\rcr(n)$, $h(n)$, and $E_{\leq k}(n) $ denote the minimum of $\crg(P)$, the maximum of $h(P)$, and the minimum of $E_{\leq k}(P) $, respectively, over all sets $P$ of $n$ points in general position in the plane. We show that the previously best known lower bound on $E_{\leq k}(n)$ is tight for $k<\lceil (4n-2) /9\rceil $ and improve it for all $k\geq \lceil (4n-2) /9 \rceil $. This in turn improves the lower bound on $\rcr(n)$ from $0.37968\binom{n} {4}+Θ(n^{3})$ to {277/729}\binom{n}{4}+Θ(n^{3})\geq 0.37997\binom{n}{4}+Θ(n^{3})$. We also give the exact values of $\rcr(n)$ and $h(n) $ for all $n\leq27$. Exact values were known only for $n\leq18$ and odd $n\leq21$ for the crossing number, and for $n\leq14$ and odd $n\leq21$ for halving lines.

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arXiv2011

On the maximum number of isosceles right triangles in a finite point set

Let $Q$ be a finite set of points in the plane. For any set $P$ of points in the plane, $S_{Q}(P)$ denotes the number of similar copies of $Q$ contained in $P$. For a fixed $n$, Erdős and Purdy asked to determine the maximum possible value of $S_{Q}(P)$, denoted by $S_{Q}(n)$, over all sets $P$ of $n$ points in the plane. We consider this problem when $Q=\triangle$ is the set of vertices of an isosceles right triangle. We give exact solutions when $n\leq9$, and provide new upper and lower bounds for $S_{\triangle}(n)$.

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arXiv2009

Sum of squares of degrees in a graph

Let $\G(v,e)$ be the set of all simple graphs with $v$ vertices and $e$ edges and let $P_2(G)=\sum d_i^2$ denote the sum of the squares of the degrees, $d_1, >..., d_v$, of the vertices of $G$. It is known that the maximum value of $P_2(G)$ for $G \in \G(v,e)$ occurs at one or both of two special graphs in $\G(v,e)$--the \qs graph or the \qc graph. For each pair $(v,e)$, we determine which of these two graphs has the larger value of $P_2(G)$. We also determine all pairs $(v,e)$ for which the values of $P_2(G)$ are the same for the \qs and the \qc graph. In addition to the \qs and \qc graphs, we find all other graphs in $\G(v,e)$ for which the maximum value of $P_2(G)$ is attained. Density questions posed by previous authors are examined.

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arXiv2009

Point-sets in general position with many similar copies of a pattern

For every pattern $P$, consisting of a finite set of points in the plane, $S_{P}(n,m)$ is defined as the largest number of similar copies of $P$ among sets of $n$ points in the plane without $m$ points on a line. A general construction, based on iterated Minkovski sums, is used to obtain new lower bounds for $S_{P}(n,m)$ when $P$ is an arbitrary pattern. Improved bounds are obtained when $P$ is a triangle or a regular polygon with few sides. It is also shown that $S_{P}(n,m)\geq n^{2-ε}$ whenever $m(n)\to \infty$ as $n \to\infty$. Finite sets with no collinear triples and not containing the 4 vertices of any parallelogram are called \emph{parallelogram-free}. The more restricted function $S_{P} ^{\nparallel}(n)$, defined as the maximum number of similar copies of $P$ among parallelogram-free sets of $n$ points, is also studied. It is proved that $Ω(n\log n)\leq S_{P}^{\nparallel}(n)\leq O(n^{3/2})$.

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arXiv2009

An inequality for Kruskal-Macaulay functions

Given integers $k\geq1$ and $n\geq0$, there is a unique way of writing $n$ as $n=\binom{n_{k}}{k}+\binom{n_{k-1}}{k-1}+...+\binom{n_{1}}{1}$ so that $0\leq n_{1}<...<n_{k-1}<n_{k}$. Using this representation, the \emph{Kruskal-Macaulay function of}$n$ is defined as $\partial^{k}(n) =\binom{n_{k}-1}{k-1}+\binom{n_{k-1}-1}{k-2}+...+\binom{n_{1}-1}% {0}.$ We show that if $a\geq0$ and $a<\partial^{k+1}(n) $, then $\partial^{k}(a) +\partial^{k+1}(n-a) \geq \partial^{k+1}(n) .$ As a corollary, we obtain a short proof of Macaulay's Theorem. Other previously known results are obtained as direct consequences.

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