🤖 AI Summary
This work investigates graph coloring problems under parity constraints, providing a unified framework that captures the computational complexity of various classical coloring variants. It introduces a general model that encompasses existing formulations by imposing combinatorial constraints—such as parity (odd/even), positivity, or zero conditions—on the number of same-colored or differently colored neighbors in vertex colorings. Employing techniques from computational complexity theory, reductions, and graph theory, the study systematically delineates the complexity landscape of all possible constraint combinations. The paper fully characterizes each case as either polynomial-time solvable (in P) or NP-complete, thereby establishing a complete complexity classification within the proposed framework.
📝 Abstract
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.