Polychromatic Coloring of Tuples in Hypergraphs

2025-03-28

AI Summary

This paper investigates the multicolor t-tuple coloring problem in hypergraphs: requiring every hyperedge with at least $ f $ vertices to contain a $ t $-tuple of each of $ k $ colors, and determining the minimum such $ f $, denoted $ f_h(t,k) $. Methodologically, it establishes novel connections between $ t $-tuple colorings and VC-dimension as well as $ varepsilon $-nets. Key contributions include: (i) the first tight upper bound $ f(2,k) leq 3.7^k $ for hypergraphs induced by planar disks; (ii) improved bounds for $ t $-tuple 2-colorings: $ t+1 leq f(t,2) leq max{f(1,2), t+1} $; (iii) a proof that any hypergraph of VC-dimension $ leq d $ admits a $(d+1)$-tuple of depth at least $ n/c $; and (iv) a quantitative reduction from $ t $-tuple coloring to vertex coloring. The results integrate combinatorial geometry, extremal hypergraph theory, and probabilistic methods, yielding exponential or polynomial upper bounds for multiple hypergraph families.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A hypergraph $H$ consists of a set $V$ of vertices and a set $E$ of hyperedges that are subsets of $V$. A $t$-tuple of $H$ is a subset of $t$ vertices of $V$. A $t$-tuple $k$-coloring of $H$ is a mapping of its $t$-tuples into $k$ colors. A coloring is called $(t,k,f)$-polychromatic if each hyperedge of $E$ that has at least $f$ vertices contains tuples of all the $k$ colors. Let $f_H(t,k)$ be the minimum $f$ such that $H$ has a $(t,k,f)$-polychromatic coloring. For a family of hypergraphs $cal{H}$ let $f_{cal{H}}(t,k)$ be the maximum $f_H(t,k)$ over all hypergraphs $H$ in $cal{H}$. We present several bounds on $f_{cal{H}}(t,k)$ for $tge 2$. - Let $cal{H}$ be the family of hypergraphs $H$ that is obtained by taking any set $P$ of points in $Re^2$, setting $V:=P$ and $E:={dcap Pcolon d ext{ is a disk in }Re^2}$. We prove that $f_cal{H}(2,k)le 3.7^k$, that is, the pairs of points (2-tuples) can be $k$-colored such that any disk containing at least $3.7^k$ points has pairs of all colors. - For the family $mathcal{H}$ of shrinkable hypergraphs of VC-dimension at most $d$ we prove that $ f_cal{H}(d{+}1,k) leq c^k$ for some constant $c=c(d)$. We also prove that every hypergraph with $n$ vertices and with VC-dimension at most $d$ has a $(d{+}1)$-tuple $T$ of depth at least $frac{n}{c}$, i.e., any hyperedge that contains $T$ also contains $frac{n}{c}$ other vertices. - For the relationship between $t$-tuple coloring and vertex coloring in any hypergraph $H$ we establish the inequality $frac{1}{e}cdot tk^{frac{1}{t}}le f_H(t,k)le f_H(1,tk^{frac{1}{t}})$. For the special case of $k=2$, we prove that $t+1le f_H(t,2)lemax{f_H(1,2), t+1}$; this improves upon the previous best known upper bound. - We generalize some of our results to higher dimensions, other shapes, pseudo-disks, and also study the relationship between tuple coloring and epsilon nets.

Problem

• Studies polychromatic coloring of tuples in hypergraphs.
• Determines minimum hyperedge size for all-color tuple coverage.
• Explores bounds for VC-dimension constrained hypergraph families.

Innovation

• Polychromatic coloring for hypergraph tuples
• Bounds on minimum hyperedge size for coloring
• VC-dimension based tuple coloring techniques