Elsevier

Discrete Mathematics

Volume 309, Issue 12, 28 June 2009, Pages 4130-4136
Discrete Mathematics

A conjecture on strong magic labelings of 2-regular graphs

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Abstract

Let sC3 denote the disjoint union of s copies of C3. For each integer t2 it is shown that the disjoint union C5(2t)C3 has a strong vertex-magic total labeling (and therefore it must also have a strong edge-magic total labeling). For each integer t3 it is shown that the disjoint union C4(2t1)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 t1 the disjoint union C7(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 C3. The results add further weight to a conjecture of MacDougall regarding the existence of vertex-magic total labeling for regular graphs.

Keywords

Labeling
Strong vertex-magic
Strong edge-magic
Super vertex-magic
2-regular

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