Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.24631

Random Tournaments Are Asymptotically Almost Surely Link-irregular

Lily Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

Bastien and Khormali (2025) conjectured that a nontrivial link-irregular tournament with $n$ vertices exists if and only if $n \ge 6$. Their work establishes the conjecture for $n \le 100$. We present a probabilistic argument for the existence of link-irregular tournaments with $n$ vertices when $n \ge 9$ to resolve the conjecture. In fact, we show that a random tournament is asymptotically almost surely link-irregular.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lily Zhang. 2026-10-17. Random Tournaments Are Asymptotically Almost Surely Link-irregular. https://arxiv.org/abs/2610.24631

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Two Erdos-Hajnal-type theorems for forbidden order-size pairs

The celebrated Erdős-Hajnal conjecture says that any graph without a fixed induced subgraph $H$ contains a very large homogeneous set. A direct analog of this conjecture is not true for hypergraphs. In this paper we present two natural variants of this problem which do hold for hypergraphs. We show that for every $r \geq 3$, $m \geq m_0(r)$ and $0 \leq f \leq \binom{m}{r}$, if an $r$-graph $G$ does not contain $m$ vertices spanning exactly $f$ edges, then $G$ contains much larger homogeneous sets than what is guaranteed to exist in general $r$-graphs. We also prove that if a $3$-graph $G$ does not contain homogeneous sets of polynomial size, then for every $m \geq 3$ there are $Ω(m^3)$ values of $f$ such that $G$ contains $m$ vertices spanning exactly $f$ edges. This makes progress on a problem of Axenovich, Bradač, Gishboliner, Mubayi and Weber.

math.CO↗
arXiv2026

Unexpectedly, a symmetry on unlabeled graphs

We exhibit the joint symmetric distribution of the following two parameters on the set of unlabeled, simple, connected graphs with $n$ vertices. The first parameter is the maximal number of leaves attached to a vertex. The second parameter is the size of the largest set of vertices sharing the same closed neighborhood minus $1$. Apparently, this is the first example of a natural, non-trivial equidistribution of graph parameters on unlabeled connected graphs on a fixed set of vertices. Our proof is enumerative, using the theory of species. Exhibiting an explicit bijection interchanging the two parameters remains an open problem.

math.CO↗
arXiv2026

Cycle lengths in the percolated hypercube

Let $Q^d_p$ be the random subgraph of the $d$-dimensional binary hypercube obtained after edge-percolation with probability $p$. It was shown recently by the authors that, for every $\varepsilon > 0$, there is some $c = c(\varepsilon)>0$ such that, if $pd\ge c$, then typically $Q^d_p$ contains a cycle of length at least $(1-\varepsilon)2^d$. We strengthen this result to show that, under the same assumptions, typically $Q^d_p$ contains cycles of all even lengths between $4$ and $(1-\varepsilon)2^d$.

math.CO↗