Monochromatic triangle-tilings in dense graphs without large independent sets

TL;DR

This paper establishes asymptotically optimal degree conditions for the existence of large weak monochromatic triangle-tilings in dense 2-edge-colored graphs with small independence number, extending classical and recent results.

Contribution

It combines previous theorems to determine degree thresholds for weak monochromatic triangle-tilings in graphs with small independence number, using the regularity lemma.

Findings

Provides asymptotically optimal bounds for triangle-tilings

Extends classical results to 2-edge-colored graphs

Uses the degree form regularity lemma in proof

Abstract

Given two graphs $H$ and $G$, an $H$-tiling is a family of vertex-disjoint copies of $H$ in $G$. A perfect $H$-tiling covers all vertices of $G$. The Corradi-Hajnal theorem (1963) states that an $n$-vertex graph $G$ with minimum degree $\delta(G)\ge 2n/3$ contains a perfect triangle-tiling. For an $n$-vertex graph $G$ with independence number $\alpha(G)=o(n)$, Balogh, Molla and Sharifzadeh (Random Structures & Algorithms, 2016) showed that a minimum degree of $(\frac12+o(1))n$ forces a perfect triangle-tiling. In a 2-edge-colored graph, Balogh, Freschi, Treglown (European J. Combin. 2026) determined the (asymptotic) minimum degree threshold for forcing a strong or weak monochromatic triangle-tiling covering a prescribed proportion of the vertices: a strong tiling requires all triangles to be in the same color class, while a weak tiling only requires each triangle to be monochromatic. In this paper, we combine the conditions from these two lines of work and prove that every $2$-edge-colored $n$-vertex graph $G$ with $\alpha(G)=o(n)$ contains a weak monochromatic triangle-tiling $\Gamma$ of size \[ |\Gamma|\ge \begin{cases} 2\delta(G)-n-o(n), & \text{if }\frac12 n\le \delta(G)\le \frac35 n,\\[2mm] \delta(G)/3-o(n), & \text{if }\delta(G)>\frac35 n. \end{cases} \] Both bounds are asymptotically optimal. We use the degree form regularity lemma in our proof.