EQUITABLE LABELINGS OF CYCLES

Every labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge (x, y) is the absolute value of the difference of the labels of x and y. By analogy with graceful labelings, we say that a labeling of the vertices of a graph of order n is minimally k-equitable if the vertices are labeled with 1,2,..., n and in the induced labeling of its edges every label either occurs exactly k times or does not occur at all. Bloom [3] posed the following question: Is the condition that k is a proper divisor of n sufficient for the cycle C(n) to have a minimal k-equitable labeling? We give a positive answer to this question. (C) 1993 John Wiley & Sons, Inc.