Two conjectures on vertex-disjoint rainbow triangles

In 1963, Dirac proved that every n-vertex graph has k vertex-disjoint triangles if n 3k and minimum degree δ (G) n+k2. 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 3k, every edge-colored graph G of order n with δᶜ (G) n+k2 contains k vertex-disjoint rainbow triangles. In another direction, Wu et al. conjectured an exact formula for anti-Ramsey number ar (n, kC₃), 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 17k5. 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 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.