Minimal \(4\)-Equitability of \(C_{2n}OK_1\)

Abstract

Every labeling of the vertices of a graph with distinct natural numbers induces a natural labeling of its edges: the label of an edge \(ae\) is the absolute value of the difference of the labels of \(a\) and \(e\). A labeling of the vertices of a graph of order \(p\) is minimally \(k\)-equitable if the vertices are labeled with elements of \({1,2, \ldots, p}\) and in the induced labeling of its edges, every label either occurs exactly \(k\) times or does not occur at all. We prove that the corona graph \(C_{2n}OK_1\) is minimally \(4\)-equitable.