BALANCED GRAPHS WITH MINIMUM DEGREE CONSTRAINTS

Let G be a finite simple graph on n vertices with minimum degree-delta = delta(G) (n = delta (mod 2)). Suppose that 0 less-than-or-equal-to delta less-than-or-equal-to n - 2, 0 less-than-or-equal-to i less-than-or-equal-to [1/2-delta]. A partition (X, Y) of V(G) is said to be an (i, delta)-partition of G if: (i) absolute value of X = [1/2n] + i, absolute value of Y = [1/2n] - i, (ii) delta([X]) greater-than-or-equal-to [1/2-delta] + i, delta([Y]) greater-than-or-equal-to [1/2-delta] - i. We prove that if G is connected then G possesses an (i, delta)-partition for some i, 0 less-than-or-equal-to i less-than-or-equal-to [1/2-delta] - 1. We show that this result is sharp and provide a family of counterexamples to Conjecture 5 in Sheehan (1988).