Almost colour-balanced spanning forests in complete graphs

Given $K_n$ whose edges are coloured red and blue, and a forest $F$ of order $n$, we seek embeddings of $F$ with small imbalance, that is, difference between the numbers of red and blue edges. We show that if the $2$-colouring of the edges of $K_n$ is balanced, meaning that the numbers of red and blue edges are equal, and $F$ has maximum degree $\Delta$, then one can find an embedding of $F$ into $K_n$ whose imbalance is at most $\Delta/2 + 18$, which is essentially best possible and resolves a conjecture of Mohr, Pardey, and Rautenbach. Furthermore, we give a tighter bound for the imbalance for small values of $\Delta$. In particular, we prove that the imbalance can be taken to be constant in the case where $\Delta<n(1/4 - \eta)$ for any constant $\eta>0$.