Strong vertex-magic and edge-magic labelings of 2-regular graphs of odd order using Kotzig completion

Let G be a 2-regular graph with 2m+1 vertices and assume that G has a strong vertex-magic total labeling. It is shown that the four graphs G∪2mC3, G∪(2m+2)C3, G∪mC8 and G∪(m+1)C8 also have a strong vertex-magic total labeling. These theorems follow from a new use of carefully prescribed Kotzig arrays. To illustrate the power of this technique, we show how just three of these arrays, combined with known labelings for smaller 2-regular graphs, immediately provide strong vertex-magic total labelings for 68 different 2-regular graphs of order 49.