On uniformly balanced graphs

Let G = (V, E) be a simple graph. A vertex labeling f : V → {0, 1} induces a partial edge labeling f ∗ : E → {0, 1} defined by f ∗(uv) = f(u) 84 KWONG, LEE, LO, SU AND WANG if and only if f(u) = f(v). For i = 0, 1, let vf(i) = |{v ∈ V : f(v) = i}|, and ef (i) = |{e ∈ E : f ∗(e) = i}|. A graph G is uniformly balanced if |ef (0)−ef (1)| ≤ 1 for any vertex labeling f that satisfies |vf(0)−vf (1)| ≤ 1. In this paper, we characterize uniformly balanced graphs, and present several ways to construct uniformly balanced graphs. In Memory of Professor F.T. Boesch