Two conjectures on vertex-disjoint rainbow triangles
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| Format: | Preprint |
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2025
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| _version_ | 1866918153086304256 |
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| author | Liu, Xu Ning, Bo Tian, Yuting |
| author_facet | Liu, Xu Ning, Bo Tian, Yuting |
| contents | In 1963, Dirac proved that every $n$-vertex graph has $k$ vertex-disjoint triangles if $n\geq 3k$ and minimum degree $δ(G)\geq \frac{n+k}{2}$. The base case $n=3k$ can be reduced to the Corrádi-Hajnál Theorem. Towards a rainbow version of Dirac's Theorem, Hu, Li, and Yang conjectured that for all positive integers $n$ and $k$ with $n\geq 3k$, every edge-colored graph $G$ of order $n$ with $δ^c(G)\geq \frac{n+k}{2}$ contains $k$ vertex-disjoint rainbow triangles. In another direction, Wu et al. conjectured an exact formula for anti-Ramsey number $ar(n,kC_3)$, generalizing the earlier work of Erdős, Sós and Simonovits. The conjecture of Hu, Li, and Yang was confirmed for the cases $k=1$ and $k=2$. However, Lo and Williams disproved the conjecture when $n\leq \frac{17k}{5}.$ It is therefore natural to ask whether the conjecture holds for $n=Ω(k)$.
In this paper, we confirm this by showing that the Hu-Li-Yang conjecture holds when $n\ge 42.5k+48$. We disprove the conjecture of Wu et al. and propose a modified conjecture. This conjecture is motivated by previous works due to Allen, Böttcher, Hladký, and Piguet on Turán number of vertex-disjoint triangles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_01880 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Two conjectures on vertex-disjoint rainbow triangles Liu, Xu Ning, Bo Tian, Yuting Combinatorics In 1963, Dirac proved that every $n$-vertex graph has $k$ vertex-disjoint triangles if $n\geq 3k$ and minimum degree $δ(G)\geq \frac{n+k}{2}$. The base case $n=3k$ can be reduced to the Corrádi-Hajnál Theorem. Towards a rainbow version of Dirac's Theorem, Hu, Li, and Yang conjectured that for all positive integers $n$ and $k$ with $n\geq 3k$, every edge-colored graph $G$ of order $n$ with $δ^c(G)\geq \frac{n+k}{2}$ contains $k$ vertex-disjoint rainbow triangles. In another direction, Wu et al. conjectured an exact formula for anti-Ramsey number $ar(n,kC_3)$, generalizing the earlier work of Erdős, Sós and Simonovits. The conjecture of Hu, Li, and Yang was confirmed for the cases $k=1$ and $k=2$. However, Lo and Williams disproved the conjecture when $n\leq \frac{17k}{5}.$ It is therefore natural to ask whether the conjecture holds for $n=Ω(k)$. In this paper, we confirm this by showing that the Hu-Li-Yang conjecture holds when $n\ge 42.5k+48$. We disprove the conjecture of Wu et al. and propose a modified conjecture. This conjecture is motivated by previous works due to Allen, Böttcher, Hladký, and Piguet on Turán number of vertex-disjoint triangles. |
| title | Two conjectures on vertex-disjoint rainbow triangles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.01880 |