Minimal k-equitability of C2n ⊙ K1, k = 2, 2n and associated graphs

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 uv is the absolute value of the difference of the labels of u and v. A labeling of the vertices of a graph of order p is minimally k-equitable if the vertices are labeled with 1, 2, ⋯, 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 graphs C_2n ⊙ K₁ are minimally k-equitable for k = 2, 2n and that C_2n+1 ⊙ K₁ are minimally (2n + 1)-equitable. Further we establish the minimal 2-equitability of graphs that are obtained by removing any set of rays from a certain part of the corona graphs C_2n ⊙ K₁.