Flip colouring of graphs II

We give results concerning two problems on the recently introduced flip colourings of graphs, a new class of local v. global phenomena. We prove that for \((b, r)\)-flip sequences with \(4 \leq b < r < b + 2 \left\lfloor\frac{b+2}{6}\right\rfloor^2\), small constructions of \((b,r)\)-flip graphs on \(O(b+r)\) vertices are possible. Furthermore, we prove that there exists \(k\)-flip sequences \((a_1, \dots, a_k)\) where \(k > 4\), such that \(a_k\) can be arbitrarily large whilst \(a_i\) is constant for \(1 \leq i < \frac{k}{4}\).