Given a graph $G$, a vertex-colouring $\sigma$ of $G$, and a subset $X\subseteq V(G)$, a colour $x \in \sigma(X)$ is said to be \emph{odd} for $X$ in $\sigma$ if it has an odd number of occurrences in $X$. We say that $\sigma$ is an \emph{odd colouring} of $G$ if it is proper and every (open) neighbourhood has an odd colour in $\sigma$. The odd chromatic number of a graph $G$, denoted by $\chi_o(G)$, is the minimum $k\in\mathbb{N}$ such that an odd colouring $\sigma \colon V(G)\to [k]$ exists. In a recent paper, Caro, Petru\v sevski and \v Skrekovski conjectured that every connected graph of maximum degree $\Delta\ge 3$ has odd-chromatic number at most $\Delta+1$. We prove that this conjecture holds asymptotically: for every connected graph $G$ with maximum degree $\Delta$, $\chi_o(G)\le\Delta+O(\ln\Delta)$ as $\Delta \to \infty$. We also prove that $\chi_o(G)\le\lfloor3\Delta/2\rfloor+2$ for every $\Delta$. If moreover the minimum degree $\delta$ of $G$ is sufficiently large, we have $\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta)$ and $\chi_o(G) = O(\chi(G)\ln \Delta)$. 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 $\min \{\deg(v),h\}$ odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant.

翻译：给定图$G$，其顶点着色$\sigma$及子集$X\subseteq V(G)$，若颜色$x \in \sigma(X)$在$X$中出现奇数次，则称该颜色对于$X$在$\sigma$下是\emph{奇}的。若$\sigma$是正常着色且每个（开）邻域在$\sigma$中均存在奇颜色，则称$\sigma$为$G$的\emph{奇着色}。图$G$的奇色数记为$\chi_o(G)$，表示存在奇着色$\sigma \colon V(G)\to [k]$的最小整数$k\in\mathbb{N}$。近期，Caro、Petru\v sevski 与 \v Skrekovski 猜想：每个最大度$\Delta\ge 3$的连通图满足奇色数至多为$\Delta+1$。我们证明该猜想渐近成立：对于任意最大度为$\Delta$的连通图$G$，当$\Delta\to\infty$时，有$\chi_o(G)\le\Delta+O(\ln\Delta)$。同时证明对任意$\Delta$，$\chi_o(G)\le\lfloor3\Delta/2\rfloor+2$。进一步地，若$G$的最小度$\delta$足够大，则$\chi_o(G) \le \chi(G) + O(\Delta \ln \Delta/\delta)$ 且 $\chi_o(G) = O(\chi(G)\ln \Delta)$。最后，对给定整数$h\ge 1$，我们将上述结果推广至$h$阶奇着色，要求每个顶点$v$的邻域中至少有$\min \{\deg(v),h\}$种奇颜色。我们的多数结果在乘法常数意义下是紧的。