New bounds for odd colourings of graphs
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Given a graph G, a vertex-colouring σ of G, and a subset X⊆ V(G), a colour x ∈σ(X) is said to be odd for X in σ if it has an odd number of occurrences in X. We say that σ is an odd colouring of G if it is proper and every (open) neighbourhood has an odd colour in σ. The odd chromatic number of a graph G, denoted by χ_o(G), is the minimum k∈ℕ such that an odd colouring σ V(G)→ [k] exists. In a recent paper, Caro, Petruševski and Škrekovski conjectured that every connected graph of maximum degree Δ≥ 3 has odd-chromatic number at most Δ+1. We prove that this conjecture holds asymptotically: for every connected graph G with maximum degree Δ, χ_o(G)≤Δ+O(lnΔ) as Δ→∞. We also prove that χ_o(G)≤⌊3Δ/2⌋+2 for every Δ. If moreover the minimum degree δ of G is sufficiently large, we have χ_o(G) ≤χ(G) + O(ΔlnΔ/δ) and χ_o(G) = O(χ(G)lnΔ). Finally, given an integer h≥ 1, we study the generalisation of these results to h-odd colourings, where every vertex v must have at least min{(v),h} odd colours in its neighbourhood. Many of our results are tight up to some multiplicative constant.
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