Mutation of Vertex-Magic Regular Graphs

A mutation of a vertex-magic total labeling of a graph \( G \) is a swap of some set of edges incident on one vertex of \( G \) with some set of edges incident with another vertex where the labels on the two sets have the same sum. Mutation has previously been seen to be a useful method for producing new labelings from old. In this paper, we study mutations which mutate labelings of regular graphs into labelings of other regular graphs. We present results of extensive computations which confirm how prolific this procedure is. These computations add weight to MacDougall’s conjecture that all non-trivial regular graphs are vertex-magic.