A cycle-magic labeling and an edge-antimagic labeling of the grid

Abstract

A total labeling of a graph G is an assignment of integers 1, 2, | V(G)| + |E(G)| to vertices and edges of G. In this paper, we present two results on the total labeling of the grid P _m□P _n (m, n ≥ 2). The first result is about magic total labeling (a notion involving constant sum). We prove that P _m□P _n (m, n ≥ 2) is C ₄-supermagic. This settles an open problem proposed by Ngurah, Salman and Susilowati in [H-supermagic labelings of graphs, Discrete Math. 310(2010)]. The second result is about antimagic total labeling (a notion involving distinct sums). We prove that the P _m□P _n(m, n ≥ 2) is (2mn +2, 1)-super-edge-antimagic total.