On friendly index sets of 2-regular graphs

Let GG be a graph with vertex set VV and edge set EE, and let AA be an abelian group. A labeling f:V→Af:V→A induces an edge labeling f∗:E→Af∗:E→A defined by f∗(xy)=f(x)+f(y)f∗(xy)=f(x)+f(y). For i∈Ai∈A, let vf(i)=card{v∈V:f(v)=i}vf(i)=card{v∈V:f(v)=i} and ef(i)=card{e∈E:f∗(e)=i}ef(i)=card{e∈E:f∗(e)=i}. A labeling ff is said to be AA-friendly if |vf(i)−vf(j)|≤1|vf(i)−vf(j)|≤1 for all (i,j)∈A×A(i,j)∈A×A, and AA-cordial if we also have |ef(i)−ef(j)|≤1|ef(i)−ef(j)|≤1 for all (i,j)∈A×A(i,j)∈A×A. When A=Z2A=Z2, the friendly index set of the graph GG is defined as {|ef(1)−ef(0)|:the vertex labelingf is Z2-friendly}{|ef(1)−ef(0)|:the vertex labelingf is Z2-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 nn is cordial if and only if n≢2n≢2 (mod 4).