On some graph-cordial Abelian groups

Abstract

Hovey introduced A-cordial labelings as a generalization of cordial and harmonious labelings [11]. 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.In the literature mostly a cordial labeling in cyclic groups is studied. Patrias and Pechenik studied the larger class of finite Abelian groups A. They posed a conjecture that for every group A there is an A-cordial labeling for almost every path. In this paper we solve this conjecture. Moreover we show that all cycle graphs are A-cordial for any Abelian group A of odd order.