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 i ∈ A, let v f (i) = card{v ∈ V : f (v) = i} and e f (i) = card{e ∈ 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) ∈ A× A, and A-cordial if we also have |e f (i) − e f ( j)| ≤ 1 for all (i, j) ∈ A × A. When A = Z2, the friendly index set of the graph G is defined as {|e f (1)− e f (0)| : the vertex labeling f 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 n is cordial if and only if n 6≡ 2 (mod 4). c © 2007 Elsevier B.V. All rights reserved.