A nearly complete complexity map of parity-constrained graph colourings: for two, three, and four-plus colours, almost every constraint combination is NP-complete, with ∨⋆ for q≥3 left open.
Colorings with neighborhood parity condition
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
In this short paper, we introduce a new vertex coloring whose motivation comes from our series on odd edge-colorings of graphs. A proper vertex coloring $\varphi$ of graph $G$ is said to be odd if for each non-isolated vertex $x\in V(G)$ there exists a color $c$ such that $\varphi^{-1}(c)\cap N(x)$ is odd-sized. We prove that every simple planar graph admits an odd $9$-coloring, and conjecture that $5$ colors always suffice.
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Complexity Classification of Colouring Problems with Parity Constraints
A nearly complete complexity map of parity-constrained graph colourings: for two, three, and four-plus colours, almost every constraint combination is NP-complete, with ∨⋆ for q≥3 left open.