SAT and IP based algorithms for magic labeling including a complete search for total magic labelings

In this paper a labeling of a graph with n vertices and m   edges is a one-to-one mapping from the union of the set of vertices and edges onto the set {1,2,…,m+n}{1,2,…,m+n}. Such a labeling is defined as magic, if one or both of the following two conditions is fulfilled: the sum of an edge label and the labels of its endpoint vertices is constant for all edges; the sum of a vertex label and the labels of its incident edges is constant for all vertices. If a graph has a labeling fulfilling both conditions, it is called a totally magic graph. We present effective IP and Sat based algorithms for the problem of finding a magic labeling for a given graph, and we extend these algorithms also to find all magic labelings for a given graph. We experimentally compare the resulted algorithms by applying it to random graphs. Finally, we demonstrate its performance by solving small cases of seven open problems within the theory of magic graphs. As main practical result we perform an exhaustive search showing that no totally magic graph with 11 vertices exists.