Multicolor and Directed Edit Distance

The editing of a combinatorial object is the alteration of some of its elements such that the resulting object satisfies a certain fixed property. The edit problem for graphs, when the edges are added or deleted, was first studied independently by the authors and Kézdy [J. Graph Theory (2008) 58(2), 123–138] and by Alon and Stav [Random Structures Algorithms (2008) 33(1), 87–104]. In this paper, a generalization of graph editing is considered for multicolorings of the complete graph as well as for directed graphs. Specifically, the number of edge-recolorings sufficient to be performed on any edge-colored complete graph to satisfy a given hereditary property is investigated. The theory for computing the edit distance is extended using random structures and so-called types or colored homomorphisms of graphs.