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 \emph{odd} for X
in σ if it has an odd number of occurrences in X. We say that σ
is an \emph{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 χo(G), is the minimum k∈N such that an
odd colouring σ:V(G)→[k] exists. In a recent paper, Caro,
Petru\v sevski and \v Skrekovski 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, χo(G)≤Δ+O(lnΔ) as Δ→∞. We also prove that χo(G)≤⌊3Δ/2⌋+2 for every
Δ. If moreover the minimum degree δ of G is sufficiently large,
we have χo(G)≤χ(G)+O(ΔlnΔ/δ) and χo(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{deg(v),h} odd colours in its neighbourhood. Many
of our results are tight up to some multiplicative constant