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Yuting Tian

Publications and source records attributed to Yuting Tian.

4 recordsLinked to original sources
arXiv2026

Rainbow triangles in edge-colored graphs with large minimum color degree

Let $G$ be an edge-colored graph on $n$ vertices, and let $\deltac(G)$ denote its minimum color degree. Li and, independently Li, Ning, Xu, and Zhang, proved that every edge-colored graph on $n$ vertices with $\deltac(G) \ge \frac{n+1}{2}$ contains a rainbow triangle. Let $\rt(G)$ denote the number of rainbow triangles in $G$, and define \[ f(n) = \min\{ \rt(G) : |V(G)| = n,\ \deltac(G) \ge (n+1)/2 \}. \] In \cite{LiNingShiZhang2024}, the following open problem was posed: determine all the values of $f(n)$. In this paper, we determine $f(n)$ completely: $f(n) = (n^2-1)/8$ for odd $n\geq 3$, $f(n) = \frac{n^2}{4} - 1$ for all even $n \ge 6,$ and $f(4) = 4$. This resolves an open problem raised in \cite{LiNingShiZhang2024}.

math.CO↗
arXiv2025

Two conjectures on vertex-disjoint rainbow triangles

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.

math.CO↗
arXiv2024

The minimum number of maximal independent sets in graphs with given order and independence number

Let $MIS(G)$ be the set of all maximal independent sets in a graph $G$, and let $mis(G)=|MIS(G)|$. In this paper, we show that for any tree $T$ with $n$ vertices and independence number $\alpha$, \[mis(T)\geq f(n-\alpha),\] and for any unicyclic graph $G$ with $n$ vertices and independence number $\alpha$, \begin{align*} mis(G)\geq \begin{cases} 2, & \text{if} \ n=4\ \text{and}\ \alpha=2, 3, & \text{if} \; \alpha=n-2 \; \text{and} \; n\neq4, 2f(n-\alpha), & \text{if} \; n\geq 5\; \text{and}\; \lceil \frac{n}{2} \rceil \leq \alpha < n-2, f(n-\alpha+2)-f(n-\alpha-3), &\text{if} \; n\geq 5, \;\text{and}\ n \; \text{is odd}, \; \text{and} \; \alpha = \lfloor \frac{n}{2} \rfloor, \end{cases} \end{align*} where $f(n)$ represent the $n$th Fibonacci number. Moreover, we also show that the above inequalities are sharp.

math.CO↗