A conjecture on strong magic labelings of 2-regular graphs

Let sC3sC3 denote the disjoint union of ss copies of C3C3. For each integer t≥2t≥2 it is shown that the disjoint union C5∪(2t)C3C5∪(2t)C3 has a strong vertex-magic total labeling (and therefore it must also have a strong edge-magic total labeling). For each integer t≥3t≥3 it is shown that the disjoint union C4∪(2t−1)C3C4∪(2t−1)C3 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≥1t≥1 the disjoint union C7∪(2t)C3C7∪(2t)C3 has a strong vertex-magic total labeling. The construction employs a technique of shifting rows of (newly constructed) Kotzig arrays to label copies of C3C3. The results add further weight to a conjecture of MacDougall regarding the existence of vertex-magic total labeling for regular graphs.