Super vertex magic chordal rings

Abstract

Let G = (V (G), E(G)) be a finite simple graph with p = |V (G)| vertices and q = |E(G)| edges, without isolated vertices or isolated edges. A vertex magic total labeling is a bijection f from V (G) ≊ E(G) to the set of consecutive integers {1, 2, · · · , p + q} with the property that, for every vertex u in V (G), the weight f (u) + ∑ uv∈E(G) f (uv) is a constant k. Moreover if f (V) = {1, 2, · · · , p}, f is called a super vertex magic total labeling. A graph is super vertex magic if it admits a super vertex magic total labeling. Very recently Javaid et al. (Labeling of Chordal Rings, Utilitas Math., 90 (2013) pp. 61-75.) studied the super vertex magic total labeling of chordal rings and raised a conjecture which claims that the chordal ring CRn(1,δ n - 1) admits such super vertex magic total labeling with magic constant 23n/4 + 2 for any odd integer δ, where 3 ≤δ ≤ n -3 and n = 0 (mod 4). In this paper we completely verify this conjecture. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.