Graph Cordiality -- Extremes and Preservers

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..