arXiv subjects

Enrique Gomez-Leos

Publications and source records attributed to Enrique Gomez-Leos.

6 recordsLinked to original sources

arXiv2025

Tiling randomly perturbed multipartite graphs

A perfect $K_r$-tiling in a graph $G$ is a collection of vertex-disjoint copies of the graph $K_r$ in $G$ that covers all vertices of $G$. In this paper, we prove that the threshold for the existence of a perfect $K_{r}$-tiling of a randomly perturbed balanced $r$-partite graph on $rn$ vertices is $n^{-2/r}$. This result is a multipartite analog of a theorem of Balogh, Treglown, and Wagner and extends our previous result, which was limited to the bipartite setting.

math.CO↗

arXiv2025

Radon-Hurwitz Grassmannian codes

Every equi-isoclinic tight fusion frame (EITFF) is a type of optimal code in a Grassmannian, consisting of subspaces of a finite-dimensional Hilbert space for which the smallest principal angle between any pair of them is as large as possible. EITFFs yield dictionaries with minimal block coherence and so are ideal for certain types of compressed sensing. By refining classical work of Lemmens and Seidel based on Radon-Hurwitz theory, we fully characterize EITFFs in the special case where the dimension of the subspaces is exactly one-half of that of the ambient space. We moreover show that each such "Radon-Hurwitz EITFF" is highly symmetric, where every even permutation is an automorphism.

cs.IT↗

arXiv2024

Tiling randomly perturbed bipartite graphs

A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that covers all vertices of $G$. Motivated by papers of Bush and Zhao and of Balogh, Treglown, and Wagner, we determine the threshold for the existence of a perfect $K_{h,h}$-tiling of a randomly perturbed bipartite graph with linear minimum degree.

math.CO↗

arXiv2024

On the proper rainbow saturation numbers of cliques, paths, and odd cycles

Given a graph $H$, we say a graph $G$ is properly rainbow $H$-saturated if there is a proper edge-coloring of $G$ which contains no rainbow copy of $H$, but adding any edge to $G$ makes such an edge-coloring impossible. The proper rainbow saturation number, denoted $\text{sat}^*(n,H)$, is the minimum number of edges in an $n$-vertex rainbow $H$-saturated graph. We determine the proper rainbow saturation number for paths up to an additive constant and asymptotically determine $\text{sat}^*(n,K_4)$. In addition, we bound $\text{sat}^*(n,H)$ when $H$ is a larger clique, tree of diameter at least 4, or odd cycle.

math.CO↗

arXiv2024

New bounds on the generalized Ramsey number $f(n,5,8)$

Let $f(n,p,q)$ denote the minimum number of colors needed to color the edges of $K_n$ so that every copy of $K_p$ receives at least $q$ distinct colors. In this note, we show $\frac{6}{7}(n-1) \leq f(n,5,8) \leq n + o(n)$. The upper bound is proven using the "conflict-free hypergraph matchings method" which was recently used by Mubayi and Joos to prove $f(n,4,5) = \frac{5}{6}n + o(n)$.

math.CO↗

arXiv2022

Orthogonal realizations of random sign patterns and other applications of the SIPP

A sign pattern is an array with entries in $\{+,-,0\}$. A matrix $Q$ is row orthogonal if $QQ^T = I$. The Strong Inner Product Property (SIPP), introduced in [B.A.~Curtis and B.L.~Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228--259, 2020], is an important tool when determining whether a sign pattern allows row orthogonality because it guarantees there is a nearby matrix with the same property, allowing zero entries to be perturbed to nonzero entries, while preserving the sign of every nonzero entry. This paper uses the SIPP to initiate the study of conditions under which random sign patterns allow row orthogonality with high probability. Building on prior work, $5\times n$ nowhere zero sign patterns that minimally allow orthogonality are determined. Conditions on zero entries in a sign pattern are established that guarantee any row orthogonal matrix with such a sign pattern has the SIPP.

math.CO↗