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Eduardo Rivera-Campo

Publications and source records attributed to Eduardo Rivera-Campo.

11 recordsLinked to original sources
arXiv2026

A note on maximal plane subgraphs of the complete twisted graph containing perfect matchings

The twisted graph $T_{n}$ is a drawing of the complete graph with $n$ vertices $v_{1},v_{2},\ldots ,v_{n}$ in which two edges $v_{i}v_{j}$ ($i<j$) and $v_{s}v_{t}$ ($s<t$) cross if and only if $i<s<t<j$ or $s<i<j<t$. We show that for any maximal plane subgraphs $S$ and $R$ of $T_{n}$, each containing at least one perfect matching, there is a sequence $S=F_0, F_1, \ldots, F_m=R$ of maximal plane subgrahs of $T_n$, also containing perfect matchings, such that for $i=0,1, \ldots, m-1$, $F_{i+1}$ can be obtained from $F_{i}$ by a single edge exchange.

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arXiv2025

Partitions of complete twisted graphs into plane spanning trees

We characterize all partitions of the complete twisted graph $T_{2n}$ into plane spanning trees. In the case of partitions of $T_{2n}$ into isomorphic plane spanning trees, we show that all trees in these partitions must be balanced double stars. As a consequence of our results, any complete topological graph with $n$ vertices contains a complete topological subgraph with $m \geq c\log^{1/8} n$ vertices that admits a partition into plane spanning trees.

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arXiv2022

Convex polygons and separation of convex sets

We prove that for any collection F of $n \ge 2$ pairwise disjoint compact convex sets in the plane there is a pair of sets A and B in F such that any line that separates A from B separates either A or B from a subcollection of F with at least n/18 sets.

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arXiv2021

On the number of heterochromatic trees in nice and beautiful colourings of complete graphs

We introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a quadratic number of different heterochromatic trees; and if the colouring is beautiful there is an exponential number of different such trees.

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arXiv2017

On contractible edges in convex decompositions

Let $Π$ be a convex decomposition of a set $P$ of $n\geq 3$ points in general position in the plane. If $Π$ consists of more than one polygon, then either $Π$ contains a deletable edge or $Π$ contains a contractible edge.

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arXiv2015

A counterexample to a result on the tree graph of a graph

Given a set of cycles C of a graph G, the tree graph of G defined by C is the graph T(G,C) whose vertices are the spanning trees of G and in which two trees R and S are adjacent if the union of R and S contains exactly one cycle and this cycle lies in C. Li et al [Discrete Math 271 (2003), 303--310] proved that if the graph T(G,C) is connected, then C cyclically spans the cycle space of G. Later, Yumei Hu [Proceedings of the 6th International Conference on Wireless Communications Networking and Mobile Computing (2010), 1--3] proved that if C is an arboreal family of cycles of G which cyclically spans the cycle space of a $2$-connected graph G, then T(G, C) is connected. In this note we present an infinite family of counterexamples to Hu's result.

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arXiv2012

Blockers for non-crossing spanning trees in complete geometric graphs

In this paper we present a complete characterization of the smallest sets that block all the simple spanning trees (SSTs) in a complete geometric graph. We also show that if a subgraph is a blocker for all SSTs of diameter at most 4, then it must block all simple spanning subgraphs, and in particular, all SSTs. For convex geometric graphs, we obtain an even stronger result: being a blocker for all SSTs of diameter at most 3 is already sufficient for blocking all simple spanning subgraphs.

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arXiv2010

On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids

The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We denote by nu(H) the number of vertices of H and by tau(H) the size of the smallest set containing at least two vertices of each hyperedge of H. For a complete geometric graph G with n > 2 vertices let H = H(G) be the hypergraph whose vertices are the edges of G and whose hyperedges are the edge sets of plane spanning trees of G. We prove that if G has at most one interior vertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) - tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given by the ground set and the bases of a matroid, respectively.

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