A conjecture on strong magic labelings of 2-regular graphs

Let s C 3 denote the disjoint union of s copies of C 3 . For each integer t ≥ 2 it is shown that the disjoint union C 5 ∪ ( 2 t ) C 3 has a strong vertex-magic total labeling (and therefore it must also have a strong edge-magic total labeling). For each integer t ≥ 3 it is shown that the disjoint union C 4 ∪ ( 2 t − 1 ) C 3 has a strong vertex-magic total labeling. These results clarify a conjecture on the magic labeling of 2-regular graphs, which posited that no such labelings existed. It is also shown that for each integer t ≥ 1 the disjoint union C 7 ∪ ( 2 t ) C 3 has a strong vertex-magic total labeling. The construction employs a technique of shifting rows of (newly constructed) Kotzig arrays to label copies of C 3 . The results add further weight to a conjecture of MacDougall regarding the existence of vertex-magic total labeling for regular graphs.