3-packings in Triangulations: Algorithms, bounds, and Complexity

2026-06-28

AI Summary

This study addresses the maximum packing problem for three-vertex subgraphs—namely, the path $P_3$, the triangle $K_3$, and the disconnected graph $P_2 \cup P_1$—in planar triangulations. By integrating extremal combinatorics, weak dual structures, and Hamiltonian cycle theory, the work develops polynomial-time algorithms and establishes structural existence conditions. The main contributions include the first tight lower bound for $P_3$-packings, $\lambda_{P_3}(G) \geq \lfloor n/5 \rfloor$; a provably efficient solution achieving $\lambda_{P_2 \cup P_1}(T) \geq \lfloor n/3 \rfloor - 2$; and a necessary and sufficient condition for the existence of triangle factors in 4-connected triangulations.

Abstract

We study $H$-packings in plane triangulations for the three-vertex graphs $H\in\{P_3,K_3,P_2\cup P_1\}$. For a graph $H$, let $λ_H(G)$ denote the maximum size of an $H$-packing in $G$, with the convention that for $H=P_2\cup P_1$ the copies are required to be induced. For $P_3$-packings, we prove that every triangulation $G$ on $n$ vertices satisfies $λ_{P_3}(G)\ge \left\lfloor \frac n5\right\rfloor$, and show that this lower bound is asymptotically tight. We also study triangle packings in triangulations and provide lower bounds for $λ_{K_3}(G)$ in terms of the maximum degree and the degree sequence. We give a face-path characterization of triangle factors in $4$-connected plane triangulations using a hamiltonian cycle and the weak duals of the two associated maximal outerplanar graphs. Finally, for induced packings by $P_2\cup P_1$, we prove that every plane triangulation $T$ on $n$ vertices satisfies $λ_{P_2\cup P_1}(T)\ge \left\lfloor \frac n3\right\rfloor-2$, and show that such a packing can be found in polynomial time.

Problem

• H-packing
• triangulation
• P3
• K3
• induced packing

Innovation

• H-packing
• plane triangulation
• induced subgraph
• polynomial-time algorithm
• triangle factor