Magic labelings of triangles

AbstractThe first proof is given that for every even integer s≥4, the graph consisting of s vertex disjoint copies of C3, (denoted sC3) is vertex-magic. Hence it is also edge-magic. It is shown that for each even integer s≥6, sC3 has vertex-magic total labelings with at least 2s−2 different magic constants. If s≡2mod4, two extra vertex-magic total labelings with the highest possible and lowest possible magic constants are given. If s=2⋅3k, k≥1, it is shown that sC3 has a vertex-magic total labeling with magic constant h if and only if (1/2)(15s+4)≤h≤(1/2)(21s+2). It is also shown that 2C3 is not vertex-magic. If s is odd, vertex-magic total labelings for sC3 with s+1 different magic constants are provided