On Hamiltonian cycles in balanced k-partite graphs

For all integers k with k≥2, if G is a balanced k-partite graph on n≥3 vertices minimum degree at least⌈n2⌉+⌊n+22⌈k+12⌉⌋−nk={⌈n2⌉+⌊n+2k+1⌋−nk:k odd n2+⌊n+2k+2⌋−nk:k even , then has Hamiltonian cycle unless 4 divides n and k∈{2,n2}. In the case where k∈{2,n2}, we can characterize graphs which do not have see that ⌈n2⌉+⌊n+22⌈k+12⌉⌋−nk+1 suffices. This result tight for k≥2 divisible by k.