Title: Two conjectures on vertex-disjoint rainbow triangles Show Chinese title

Title: 两个关于顶点不相交的彩虹三角形的猜想

Abstract: In 1963, Dirac proved that every $n$-vertex graph has $k$ vertex-disjoint triangles if $n\geq 3k$ and minimum degree $\delta(G)\geq \frac{n+k}{2}$. The base case $n=3k$ can be reduced to the Corr\'adi-Hajn\'al 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 $\delta^c(G)\geq \frac{n+k}{2}$ 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 Erd\H{o}s, S\'os 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 $n=\Omega(k)$. 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, B\"{o}ttcher, Hladk\'{y}, and Piguet on Tur\'an number of vertex-disjoint triangles. Show Chinese abstract

Abstract: 1963年，Dirac证明了每个具有$n$个顶点的图如果满足$n\geq 3k$且最小度为$\delta(G)\geq \frac{n+k}{2}$，则包含$k$个顶点不相交的三角形。基数情况$n=3k$可以归约到 Corrádi-Hajnál 定理。 朝着Dirac定理的彩虹版本，Hu、Li和Yang猜想，对于所有满足$n\geq 3k$的正整数$n$和$k$，每个阶为$n$的边着色图$G$具有$\delta^c(G)\geq \frac{n+k}{2}$，包含$k$个顶点不相交的彩虹三角形。 在另一个方向上，Wu等人 猜想了一个反Ramsey数的精确公式$ar(n,kC_3)$，推广了Erdős, Sós和Simonovits的早期工作。 Hu、Li和Yang的猜想已被证实适用于情况$k=1$和$k=2$。 然而，Lo和Williams在$n\leq \frac{17k}{5}.$时否定了该猜想。因此自然地要问该猜想是否在$n=\Omega(k)$时成立。 在本文中，我们通过证明当$n\ge 42.5k+48$时Hu-Li-Yang猜想成立来确认这一点。 我们否定了Wu等人的猜想，并提出了一个修改后的猜想。 该猜想受到Allen、Böttcher、Hladký和Piguet关于顶点不相交三角形的Turán数的先前工作的启发。