AI 中文总结
研究具有奇偶性约束的图着色变体，通过考虑顶点邻域颜色数量的多种限制条件，将许多已知着色问题纳入此框架，全面研究了涉及奇偶性的不同约束组合的计算复杂性。
AI 中文摘要
我们研究具有奇偶性约束的图着色变体。具体而言，考虑图\(G\)的\(q\)着色\(c\colon V(G)\to\{1,\dots,q\}\)，对于每个顶点\(v\in V(G)\)，与\(v\)颜色相同的邻居\(w\)的数量限制为奇数、偶数、正数、零或其组合。对于每个不同于\(c(v)\)的颜色\(i\)，与\(v\)颜色为\(i\)的邻居\(w\)的数量也受类似类型约束。许多已知着色问题可在此框架内描述，因此考虑变体是已知着色问题的自然推广。我们全面研究了涉及奇偶性的不同约束组合的计算复杂性。
英文摘要
We study variants of graph colouring with parity constraints. More specifically, we consider $q$-colourings $c\colon V(G)\rightarrow \{1,\dots,q\}$ of a graph $G$ where, for every vertex $v\in V(G)$, the number of neighbours $w$ of $v$ with $c(w)=c(v)$ is restricted to be odd, even, positive, zero or a combination thereof. For every colour $i\neq c(v)$ the number of neighbours $w$ of $v$ with $c(w)=i$ is restricted by a constraint of similar type. Many known colouring problems such as proper colouring, defective colouring, exact defective colouring, odd colouring, and strong odd colouring can be described within this framework of constraining graph colourings, and therefore considering variants constitutes a natural generalisation of known colouring problems. We provide a comprehensive study of the computational complexity of different combinations of constraints involving parity.
