DMGT

Discussiones Mathematicae Graph Theory 32(3) (2012) 557-567
DOI: https://doi.org/10.7151/dmgt.1626

Generalized Graph Cordiality

Oliver Pechenik and Jennifer Wise

Department of Mathematics
University of Illinois
Urbana, IL, 61801

Abstract

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 edge-labeling 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 K_m,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 C_k, where k ≡ 2(mod 4). All ladders P₂ ^[¯] P_k are V₄-cordial except C₄. All prisms are V₄-cordial except P₂ ^[¯] C_k, 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.

Keywords: graph labeling, cordial graph, A-cordial, quasigroup

2010 Mathematics Subject Classification: 05C78, 05C25.

References

[1]	I. Cahit, Cordial graphs: a weaker version of graceful and harmonious graphs, Ars Combin. 23 (1987) 201--207.
[2]	J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. 18 (2011) }.
[3]	R.L. Graham and N.J.A. Sloane, On additive bases and harmonious graphs, SIAM J. Algebraic Discrete Methods 1 (1980) 382--404, doi: 10.1137/0601045.
[4]	M. Hovey, {A}, Discrete Math. 93 (1991) 183--194, doi: 10.1016/0012-365X(91)90254-Y.
[5]	G. McAlexander, Undergraduate thesis, (Mary Baldwin College, c.2007).
[6]	A. Riskin, ℤ₂²-cordiality of complete and complete bipartite graphs, (http://arxiv.org/abs/0709.0290v1), September 2007.

Received 30 March 2011
Revised 30 September 2011
Accepted 30 September 2011