Improved lower bounds on parity vertex colourings of binary trees

TL;DR

This paper establishes improved lower bounds for the parity vertex chromatic number of binary trees, explores its properties, and analyzes the computational complexity of related decision problems.

Contribution

It provides new lower bounds for parity vertex colourings of binary trees, shows non-monotonicity with minors, and proves complexity results for computing the parity chromatic number.

Findings

Lower bound for binary trees: (d) + rac{1}{4} \u221a{d} - rac{1}{2}

Parity vertex chromatic number is not minor-monotone

Checking if a colouring is a parity vertex colouring is coNP-complete

Abstract

A vertex colouring is called a \emph{parity vertex colouring} if every path in $G$ contains an odd number of occurrences of some colour. Let $\chi_{p}(G)$ be the minimal number of colours in a parity vertex colouring of $G$. We show that $\chi_{p}(B^*) \ge \sqrt{d} + \frac{1}{4} \log_2(d) - \frac{1}{2}$ where $B^*$ is a subdivision of the complete binary tree $B_d$. This improves the previously known bound $\chi_{p}(B^*) \ge \sqrt{d}$ and enhances the techniques used for proving lower bounds. We use this result to show that $\chi_{p}(T) > \sqrt[3]{\log{n}}$ where $T$ is any binary tree with $n$ vertices. These lower bounds are also lower bounds for the conflict-free colouring. We also prove that $\chi_{p}(G)$ is not monotone with respect to minors and determine its value for cycles. Furthermore, we study complexity of computing the parity vertex chromatic number $\chi_{p}(G)$. We show that checking whether a vertex colouring is a parity vertex colouring is coNP-complete. Then we use Courcelle's theorem to prove that the problem of checking whether $\chi_{p}(G) \le k$ is fixed-parameter tractable with respect $k$ and the treewidth of $G$.