Balanced graphs with minimum degree constraints

Abstract

Let G be a finite simple graph on n vertices with minimum degree δ = δ(G) (n ≡ δ(mod 2)). Suppose that $\text{0 ⩽ δ ⩽ n − 2, 0 ⩽ i ⩽ ⌊}\text{1}\text{2}\text{δ⌋}$. A partition (X,Y) of V(G) is said to be an (i, δ)-partition of G if:

• (i)

  $$\text{|X|=⌈}\text{1}\text{2}\text{n⌉+i,|Y|=⌊}\text{1}\text{2}\text{n”−i}$$

• (ii)

  $$\text{δ(〈X〉)⩾⌈}\text{1}\text{2}\text{δ⌉ +i,δ(〈Y〉)⩾;⌊}\text{1}\text{2}\text{⌋ −i}$$

  .

We prove that if G is connected then G possesses an (i, δ)-partition for some i, $\text{0⩽i⩽⌊}\text{1}\text{2}\text{δ⌋ − 1}$. We show that this result is sharp and provide a family of counterexamples to Conjecture 5 in Sheehan (1988).