The total edge irregularity strength of hexagonal grid graphs

For a graph G = (V, E), a labeling partial derivative : V boolean OR E -> {1,2, . . . , k} is called an edge irregular total k-labeling of G if the weights of any two different edges are distinct, where the weight of the edge xy under partial derivative is defined to be wt(xy) = partial derivative(x) + partial derivative(xy) + partial derivative(y). The total edge irregularity strength tes(G) of G is the minimum k for which G has an edge irregular total k-labeling. Al-Mushayt et al. "prove" that tes(H-n(m)) =l3mn+2(m+n)+13m for the hexagonal grid graph H-n(m), but the labeling they constructed is actually not a total [3mn+2(m+n)+1/3]-labeling. In this paper, we first describe a correctedge irregular total [3mn+2(m+n)+1/3]-labeling of H-n(m) for any m, n >= 1, and so show that tes(H-n(m)) = [3mn+2(m+n)+1/3]. Moreover, we determine the exact value of the total edge irregularity strength for a more general hexagonal grid graph H-n(m1,m2,...,mn) by giving an edge irregular total tes(H-n(m1,m2,...,mn))-labeling, where H-n(m1,m2,...,mn) consists of ncolumns of hexagons and hasmihexagons in the i-th column, n >= 2, and m(1), . . . , m(n )>= 1.