Almost-perfect packings and Tuza's conjecture in the random geometric graph

2026-06-08

AI Summary

This study investigates the validity of Tuza’s conjecture and the problem of nearly perfect edge-disjoint packings of a fixed graph \(F\) in random geometric graphs. By integrating tools from random geometric graph theory, extremal graph theory, and probabilistic combinatorics, the work establishes— for the first time in this model—that Tuza’s conjecture holds across a broad range of edge densities and demonstrates the existence of almost perfect packings for an infinite family of graphs. Additionally, several negative results are provided, shedding light on the limiting behavior of packing and covering structures. These contributions not only extend the applicability of Tuza’s conjecture but also introduce novel techniques and perspectives for addressing extremal combinatorial problems in random geometric settings.

Abstract

The triangle packing number $ν(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2ν(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random geometric graph for a large range of densities. We also study the problem of covering almost all edges of the random geometric graph with edge-disjoint copies of some fixed graph $F$. In particular, we show the existence of almost-perfect packings for an infinite family of $F$, and state some negative results as well.

Problem

• triangle packing
• Tuza's conjecture
• random geometric graph
• almost-perfect packing
• edge-disjoint copies

Innovation

• random geometric graph
• Tuza's conjecture
• triangle packing
• almost-perfect packing
• edge-disjoint copies