Weak Monochromatic Triangle-Tiling Bounds

• Weak Monochromatic Triangle-Tiling is a tiling construction in 2-edge-coloured graphs using vertex-disjoint monochromatic triangles that may differ in color.
• The research establishes asymptotically tight bounds for the number of triangles based on minimum degree and independence number constraints.
• Methodologies include applying the Szemerédi regularity lemma and multipartite embedding techniques to unify classical tiling theory with colored graph advances.

A weak monochromatic triangle-tiling in a 2-edge-coloured graph is a construction comprising vertex-disjoint monochromatic triangles, where each triangle may be colored differently. This concept extends tiling theory in graph combinatorics to colored scenarios and interacts fundamentally with minimum degree and independence number constraints. Explicit bounds and constructions related to weak monochromatic triangle-tilings in dense graphs have been recently established, unifying classical tiling theorems with recent advances for colored graphs with small independence numbers (Hou et al., 26 Jan 2026).

1. Precise Definitions and Distinction

Given an undirected, finite, simple graph $G$ with vertex set $V(G)$ and edge set $E(G)$, together with a coloring function $\phi: E(G) \to \{\text{red}, \text{blue}\}$, the structure $(G, \phi)$ is termed a 2-edge-coloured graph. Within this framework:

• A strong monochromatic $K_3$-tiling is a set of vertex-disjoint triangles (i.e., copies of $K_3$) all of which are in the same color.
• A weak monochromatic $K_3$-tiling is a set of vertex-disjoint triangles, each monochromatic, but where triangles may occur in either color.
• The maximum size of a weak monochromatic $K_3$-tiling is denoted $\nu_w(G, K_3)$.

While strong tilings demand color uniformity, weak tilings prioritize the monochromatic property per triangle, thus broadening applicability in edge-colored settings.

2. Main Theorem: Asymptotically Tight Bounds and Conditions

Let $G$ be an $n$-vertex graph with independence number $\alpha(G) = o(n)$ and minimum degree $\delta(G)$. For any sufficiently small $\gamma > 0$, and $n \geq n_0(\gamma)$ large enough, every 2-edge-coloring of $G$ contains a weak monochromatic $K_3$-tiling $\Gamma$, with size satisfying: $$|\Gamma| \ge \begin{cases} 2\,\delta(G) - n - o(n), & \text{ if } \tfrac12 n \le \delta(G) \le \tfrac35 n, \ \delta(G)/3 - o(n), & \text{ if } \delta(G) > \tfrac35 n. \end{cases}$$ Both bounds are proven to be asymptotically optimal by matching extremal constructions (Hou et al., 26 Jan 2026).

3. Extremal Constructions Demonstrating Tightness

To show optimality, one constructs complete multipartite graphs with part sizes tailored according to the minimum degree $\delta(G)$. Consider $\ell = \lceil n/(n-\delta)\rceil$, with partition $V_1, \dots, V_\ell$ such that $|V_1| = n-\delta$ for the first $\ell-1$ parts. All edges between parts are present, and intra-part edges induce triangle-free graphs with independence number $o(n)$—via Ramsey–Turán estimates. By coloring edges from $V_1$ to the rest red and all remaining edges blue, no vertex from $V_1$ lies in a monochromatic triangle. Thus, at most $(n - |V_1|)/3 = \delta/3$ monochromatic triangles can be packed. For $\tfrac12n < \delta \le \tfrac35n$, optimizing yields an upper bound of $2\delta - n$, matching the theorem.

4. Structural and Methodological Foundations

The proof utilizes the Szemerédi regularity lemma in degree form to partition $G$ as $V(G)=V_0\cup V_1\cup\cdots\cup V_k$, where each $V_i$ (for $i\geq 1$) forms clusters of equal size $m$. A reduced graph $R$ is constructed on these clusters, encoding the regular pairs of density $\geq \beta$. For $\delta(G)\geq \frac12 n$, the reduced graph has minimum degree $$\delta(R) \geq (\delta(G)/n - (\beta+\varepsilon))k$$.

Central to the method is finding large tilings in $R$ using the graph $F_2$ (two triangles sharing a vertex). Graphs with $\frac12k < \delta(R) \leq \frac35k$ permit $F_2$-tilings of size at least $2\delta(R) - k - O(1)$, covering $5$ clusters per $F_2$. Embedded in the original graph via a five-partite blow-up lemma, each $F_2$ block yields an almost-perfect weak monochromatic triangle-tiling covering $(1-O(\sqrt\varepsilon))m$ triangles. The method iterates, accounting for the residual minimum degree, establishing tight lower bounds in both minimum degree regimes.

5. Core Intermediate Results and Innovations

Progress relies on two fundamental lemmas:

• $F_2$-tilings in dense graphs: Any $k$-vertex graph with minimum degree $\leq \frac35k$ admits an $F_2$-tiling of size at least $2\delta(R)-k-O(1)$.
• Five-partite embedding lemma: In a 5-partite graph whose pairs realize blow-ups of $F_2$ with prescribed $\varepsilon$-regular pairs of density $\geq \beta$, an almost-perfect weak monochromatic triangle-tiling can be embedded. This leverages the regularity slicing lemma, greedy domination, and intricate case analysis to circumvent large independent sets.

Notably, the proof's constant hierarchy ($$1 \gg \gamma \gg \beta \gg \varepsilon \gg 1/k_0 \gg \alpha \gg 1/n$$) is vital for ensuring the fine-grained regularity required for block tilings and controlling error terms via $o(n)$.

6. Historical and Theoretical Context

The framework for tilings in graphs originates with the Corrádi–Hajnal theorem (1963), establishing that $\delta(G)\ge2n/3$ guarantees perfect triangle-tilings in simple graphs. Subsequent advances by Balogh, Molla, and Sharifzadeh (2016) demonstrate that when $\alpha(G)=o(n)$, only $(\frac12+o(1))n$ minimum degree suffices for perfect tilings. Progress in 2-edge-colored settings by Balogh, Freschi, and Treglown (2026) further refines asymptotic minimum degree thresholds needed for strong and weak monochromatic tilings over linear fractions of vertices. The current results (Hou et al., 26 Jan 2026) unify and generalize these threads, quantifying the maximal size of weak monochromatic $K_3$-tilings under collapsed independence number conditions and coloring constraints.

7. Significance and Implications

These results comprehensively characterize structural properties and packing limits of weak monochromatic triangle-tilings in dense 2-edge-colored graphs with sublinear independence number. They complete the program of determining best-possible bounds for monochromatic tilings in colored settings when large independent sets are absent, clarifying the role of minimum degree and regular partitions in colored tiling theory. A plausible implication is further extensions to monochromatic tilings for other small graphs or higher order cliques under analogous independence and minimum degree constraints. These findings also frame the dependence of tiling parameters on chromatic and structural graph properties, reinforcing the efficacy of regularity-based methodologies in extremal combinatorics.