Magic labelings of triangles

Abstract

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