A conjecture on strong magic labelings of 2-regular graphs

摘要

Let sC(3) denote the disjoint union of s copies of C-3. For each integer t >= 2 it is shown that the disjoint union C5 boolean OR (2t)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 boolean OR (2t - 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 boolean OR (2t)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.