On friendly index sets of 2-regular graphs

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f:V->A induces an edge labeling f^*:E->A defined by f^*(xy)=f(x)+f(y). For [email protected]?A, let v"f(i)=card{[email protected]?V:f(v)=i} and e"f(i)=card{[email protected]?E:f^*(e)=i}. A labeling f is said to be A-friendly if |v"f(i)-v"f(j)|@?1 for all (i,j)@?AxA, and A-cordial if we also have |e"f(i)-e"f(j)|@?1 for all (i,j)@?AxA. When A=Z"2, the friendly index set of the graph G is defined as {|e"f(1)-e"f(0)|:the vertex labelingf is Z"2-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if [email protected]?2 (mod 4).