We study how the complexity of the graph colouring problems star colouring and restricted star colouring vary with the maximum degree of the graph. Restricted star colouring (in short, rs colouring) is a variant of star colouring. For $k\in \mathbb{N}$, a $k$-colouring of a graph $G$ is a function $f\colon V(G)\to \mathbb{Z}_k$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$. A $k$-colouring of $G$ is called a $k$-star colouring of $G$ if there is no path $u,v,w,x$ in $G$ with $f(u)=f(w)$ and $f(v)=f(x)$. A $k$-colouring of $G$ is called a $k$-rs colouring of $G$ if there is no path $u,v,w$ in $G$ with $f(v)>f(u)=f(w)$. For $k\in \mathbb{N}$, the problem $k$-STAR COLOURABILITY takes a graph $G$ as input and asks whether $G$ admits a $k$-star colouring. The problem $k$-RS COLOURABILITY is defined similarly. Recently, Brause et al. (Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with respect to the graph diameter. We study the complexity of $k$-star colouring and $k$-rs colouring with respect to the maximum degree for all $k\geq 3$. For $k\geq 3$, let us denote the least integer $d$ such that $k$-STAR COLOURABILITY (resp. $k$-RS COLOURABILITY) is NP-complete for graphs of maximum degree $d$ by $L_s^{(k)}$ (resp. $L_{rs}^{(k)}$). We prove that for $k=5$ and $k\geq 7$, $k$-STAR COLOURABILITY is NP-complete for graphs of maximum degree $k-1$. We also show that $4$-RS COLOURABILITY is NP-complete for planar 3-regular graphs of girth 5 and $k$-RS COLOURABILITY is NP-complete for triangle-free graphs of maximum degree $k-1$ for $k\geq 5$. Using these results, we prove the following: (i) for $k\geq 4$ and $d\leq k-1$, $k$-STAR COLOURABILITY is NP-complete for $d$-regular graphs if and only if $d\geq L_s^{(k)}$; and (ii) for $k\geq 4$, $k$-RS COLOURABILITY is NP-complete for $d$-regular graphs if and only if $L_{rs}^{(k)}\leq d\leq k-1$.

翻译：我们研究图着色问题——星着色与受限星着色的复杂度如何随图的最大度变化。受限星着色（简称rs着色）是星着色的一种变体。对于$k\in \mathbb{N}$，图$G$的$k$-着色是一个函数$f\colon V(G)\to \mathbb{Z}_k$，满足对$G$的每条边$uv$有$f(u)\neq f(v)$。若$G$中不存在路径$u,v,w,x$使得$f(u)=f(w)$且$f(v)=f(x)$，则称$G$的$k$-着图为$k$-星着色。若$G$中不存在路径$u,v,w$使得$f(v)>f(u)=f(w)$，则称$G$的$k$-着图为$k$-rs着色。对于$k\in \mathbb{N}$，问题$k$-星可着色性以图$G$为输入，询问$G$是否存在$k$-星着色。问题$k$-rs可着色性的定义类似。最近，Brause等人（Electron. J. Comb., 2022）研究了3-星着色关于图直径的复杂度。我们研究所有$k\geq 3$时，$k$-星着色与$k$-rs着色关于最大度的复杂度。对于$k\geq 3$，记使$k$-星可着色性（相应地，$k$-rs可着色性）对最大度为$d$的图为NP完全的的最小整数$d$为$L_s^{(k)}$（相应地，$L_{rs}^{(k)}$）。我们证明当$k=5$且$k\geq 7$时，$k$-星可着色性对最大度为$k-1$的图是NP完全的。我们还证明$4$-rs可着色性对围长为5的平面3正则图是NP完全的，且当$k\geq 5$时，$k$-rs可着色性对最大度为$k-1$的无三角形图是NP完全的。利用这些结果，我们证明以下结论：(i) 对于$k\geq 4$且$d\leq k-1$，$k$-星可着色性对$d$正则图是NP完全的当且仅当$d\geq L_s^{(k)}$；(ii) 对于$k\geq 4$，$k$-rs可着色性对$d$正则图是NP完全的当且仅当$L_{rs}^{(k)}\leq d\leq k-1$。