Monochromatic triangle-tilings in dense graphs without large independent sets
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912849869144064 |
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| author | Hou, Xinmin Wang, Xiangyang Yin, Zhi |
| author_facet | Hou, Xinmin Wang, Xiangyang Yin, Zhi |
| contents | 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 $δ(G)\ge 2n/3$ contains a perfect triangle-tiling. For an $n$-vertex graph $G$ with independence number $α(G)=o(n)$, 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 $α(G)=o(n)$ contains a weak monochromatic triangle-tiling $Γ$ of size \[ |Γ|\ge \begin{cases} 2δ(G)-n-o(n), & \text{if }\frac12 n\le δ(G)\le \frac35 n,\\[2mm] δ(G)/3-o(n), & \text{if }δ(G)>\frac35 n. \end{cases} \] Both bounds are asymptotically optimal. We use the degree form regularity lemma in our proof. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18565 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Monochromatic triangle-tilings in dense graphs without large independent sets Hou, Xinmin Wang, Xiangyang Yin, Zhi Combinatorics 05C35 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 $δ(G)\ge 2n/3$ contains a perfect triangle-tiling. For an $n$-vertex graph $G$ with independence number $α(G)=o(n)$, 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 $α(G)=o(n)$ contains a weak monochromatic triangle-tiling $Γ$ of size \[ |Γ|\ge \begin{cases} 2δ(G)-n-o(n), & \text{if }\frac12 n\le δ(G)\le \frac35 n,\\[2mm] δ(G)/3-o(n), & \text{if }δ(G)>\frac35 n. \end{cases} \] Both bounds are asymptotically optimal. We use the degree form regularity lemma in our proof. |
| title | Monochromatic triangle-tilings in dense graphs without large independent sets |
| topic | Combinatorics 05C35 |
| url | https://arxiv.org/abs/2601.18565 |