Two conjectures on vertex-disjoint rainbow triangles
Xu Liu, Bo Ning, Yuting Tian
TL;DR
The paper investigates two conjectures about rainbow triangles in edge-colored graphs. It provides a rainbow-Dirac-type result showing that, for sufficiently large $n$ relative to $k$, a graph with minimum color degree $\delta^c(G)\ge (n+k)/2$ contains $k$ vertex-disjoint rainbow triangles, proving this for $n\ge 42.5k+48$; it also engages the anti-Ramsey problem for $kC_3$ by analyzing and ultimately refuting a conjecture for $ar(n,kC_3)$ through extremal colorings that exceed the proposed bound while avoiding rainbow $kC_3$, motivating a modified conjecture. The authors develop a complex inductive framework using structures like $RF_s(v)$ (rainbow friendship graphs) and intricate counting arguments to bound rainbow triangles and reach contradictions in the disproof. Additionally, the work connects to Turán-type questions on vertex-disjoint triangles, guiding the proposed modifications and future directions in the anti-Ramsey landscape.
Abstract
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.