Generalized graph cordiality

Hovey introduced A-cordial labelings in [4] as a simultaneous generalization of cordial and harmonious labelings. If A is an abelian group, then a labeling f : V (G) → A of the vertices of some graph G induces an edgelabeling on G; the edge uv receives the label f(u) + f(v). A graph G is A-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. Research on A-cordiality has focused on the case where A is cyclic. In this paper, we investigate V4-cordiality of many families of graphs, namely complete bipartite graphs, paths, cycles, ladders, prisms, and hypercubes. We find that all complete bipartite graphs are V4-cordial except Km,n where m,n ≡ 2(mod 4). All paths are V4-cordial except P4 and P5. All cycles are V4-cordial except C4, C5, and Ck, where k ≡ 2(mod 4). All ladders P2 Pk are V4-cordial except C4. All prisms are V4-cordial except P2 Ck, where k ≡ 2(mod 4). All hypercubes are V4-cordial, except C4. Finally, we introduce a generalization of A-cordiality involving digraphs and quasigroups, and we show that there are infinitely many Q-cordial digraphs for every quasigroup Q.