Flip colouring of graphs

It is proved that for integers b , r such that 3 ≤ b < r ≤ b + 1 2 - 1 , there exists a red/blue edge-colored graph such that the red degree of every vertex is r , the blue degree of every vertex is b , yet in the closed neighbourhood of every vertex there are more blue edges than red edges. The upper bound r ≤ b + 1 2 - 1 is best possible for any b ≥ 3 . We further extend this theorem to more than two colours, and to larger neighbourhoods. A useful result required in some of our proofs, of independent interest, is that for integers r , t such that 0 ≤ t ≤ r 2 2 - 5 r 3 / 2 , there exists an r -regular graph in which each open neighbourhood induces precisely t edges. Several explicit constructions are introduced and relationships with constant linked graphs, ( r , b )-regular graphs and vertex transitive graphs are revealed.