arXiv subjects

Qiancheng Ouyang

Publications and source records attributed to Qiancheng Ouyang.

3 recordsLinked to original sources

arXiv2026

New bounds for proper $h$-conflict-free colourings

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $χ_{\rm pcf}^h(G)$ denote the minimum $k$ such that such a colouring exists. We show that for every fixed $h\ge 1$, every graph $G$ of maximum degree $Δ$ satisfies $χ_{\rm pcf}^h(G) \le hΔ+ \mathcal{O}(\log Δ)$. This expands on the work of Cho et al., and improves a recent result of Liu and Reed in the case $h=1$. We conjecture that for every $h\ge 1$ and every graph $G$ of maximum degree $Δ$ sufficiently large, the bound $χ_{\rm pcf}^h(G) \le hΔ+ 1$ should hold, which would be tight. When the minimum degree $δ$ of $G$ is sufficiently large, namely $δ\ge \max\{100h, 2000\log Δ\}$, we show that this upper bound can be further reduced to $χ_{\rm{pcf}}^h(G) \le Δ+ \mathcal{O}(\sqrt{hΔ})$. This improves a recent bound from Kamyczura and Przybyło when $δ\le \sqrt{hΔ}$.

math.CO↗

arXiv2025

An exact Ore-degree condition for Hamilton cycles in oriented graphs

An oriented graph is a digraph that contains no 2-cycles, i.e., there is at most one arc between any two vertices. We show that every oriented graph $G$ of sufficiently large order $n$ with $\mathrm{deg}^+(x) +\mathrm{deg}^{-}(y)\geq (3n-3)/4$ whenever $G$ does not have an edge from $x$ to $y$ contains a Hamilton cycle. This is best possible and solves a problem of Kühn and Osthus from 2012. Our result generalizes the result of Keevash, Kühn, and Osthus and improves the asymptotic bound obtained by Kelly, Kühn, and Osthus.

math.CO↗

arXiv2023

New bounds for odd colourings of graphs

Given a graph $G$, a vertex-colouring $σ$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in σ(X)$ is said to be \emph{odd} for $X$ in $σ$ if it has an odd number of occurrences in $X$. We say that $σ$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $σ$. The odd chromatic number of a graph $G$, denoted by $χ_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $σ\colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $Δ\ge 3$ has odd-chromatic number at most $Δ+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $Δ$, $χ_o(G)\leΔ+O(\lnΔ)$ as $Δ\to \infty$. We also prove that $χ_o(G)\le\lfloor3Δ/2\rfloor+2$ for every $Δ$. If moreover the minimum degree $δ$ of $G$ is sufficiently large, we have $χ_o(G) \le χ(G) + O(Δ\ln Δ/δ)$ and $χ_o(G) = O(χ(G)\ln Δ)$. Finally, given an integer $h\ge 1$, we study the generalisation of these results to $h$-odd colourings, where every vertex $v$ must have at least $\min \{°(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant.

math.CO↗