Improved lower bounds on parity vertex colourings of binary trees
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A vertex colouring is called a parity vertex colouring if every path in G contains an odd number of occurrences of some colour. Let χ_p(G) be the minimal number of colours in a parity vertex colouring of G. We show that χ_p(B^*) >√(d) + 1/4_2(d) - 1/2 where B^* is a subdivision of the complete binary tree B_d. This improves the previously known bound χ_p(B^*) >√(d) and enhances the techniques used for proving lower bounds. We use this result to show that χ_p(T) > √(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 χ_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 χ_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 χ_p(G) < k is fixed-parameter tractable with respect k and the treewidth of G.
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