Author

Ziyuan Zhao

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Preprint Sep 2026

Longest cycles intersect linearly in highly connected graphs

A longstanding conjecture attributed to Smith (1984) asserts that for every $k\ge2$, any two longest cycles in a $k$-connected graph share at least $k$ vertices. In this paper, we prove the first linear lower bound, showing that any two longest cycles in a $k$-connected graph share at least $k/600$ vertices. Departing...

Jie Ma, Bo Ning, Zi-Yuan Zhao · 0 citations

Preprint Oct 2026

Linear circumference in vertex-transitive graphs

We prove that there is an absolute constant $c>0$ such that every connected vertex-transitive graph $G$ on $n \ge 3$ vertices contains a cycle of length at least $cn$. Moreover, every such graph with sufficiently large degree $d$ contains a cycle of length at least $(1-d^{-1/100})n$. This gives the first linear bound t...

Jie Ma, Zi-Yuan Zhao · 0 citations

Preprint Aug 2026

An exact formula for Erd\H{o}s'problem 1005

In 1943, Erd\H{o}s considered the minimum number $f(n)$ of terms between two fractions in the Farey sequence of order $n$ whose numerators and denominators are oppositely ordered. Determining the constant $c$ in $f(n)=(c+o(1))n$ is known as Erd\H{o}s Problem 1005. Recently, Cipollini solved this asymptotic problem by p...

Yanmohan Wang, Mingxu Xie, Ziyuan Zhao · 0 citations