Preservers of Graph and Digraph Cordiality

An undirected graph is said to be cordial if there is a friendly (0,1)-labeling of the vertices that induces a friendly (0,1)-labeling of the edges. An undirected graph \(G\) is said to be \((2,3)\)-orientable if there exists a friendly (0,1)-labeling of the vertices of \(G\) such that about one-third of the edges are incident to vertices labeled the same. That is, there is some digraph that is an orientation of \(G\) that is \((2,3)\)-cordial. Examples of the smallest noncordial/non-\((2,3)\)-orientable graphs are given, and upper bounds on the possible number of edges in a cordial/\((2,3)\)-orientable graph are presented. It is also shown that if \(T\) is a linear operator on the set of all undirected graphs on \(n\) vertices that strongly preserves the set of cordial graphs or the set of \((2,3)\)-orientable graphs, then \(T\) is a vertex permutation.