Vertex-Magic Edge Labeling Games on Graphs with Cycles

Given a graph \( G \), let \( E \) be the number of edges in \( G \). A \emph{vertex-magic edge labeling} of \( G \), defined by Wallis [12] in 2001, is a one-to-one mapping from the set of edges onto the set \(\{1, 2, \ldots, E\}\) with the property that at any vertex the sum of the labels of all the edges incident to that vertex is the same constant. In 2003, Hartnell and Rall [5] introduced a two-player game based on these labelings, and proved some nice results about winning strategies on graphs that contain vertices of degree one. In this paper, we prove results about winning strategies for certain graphs with cycles where the minimum degree is two.