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 is an element of A, let v(f)(i) = card {v is an element of V : f (v) = i} and e(f)(i) = card {e is an element of E : f*(e) = i}. A labeling f is said to be A-friendly if vertical bar v(f)(i) - v(f)(j)vertical bar <= 1 for all (i, j) is an element of A x A, and A-cordial if we also have vertical bar e(f)(i) - e(f)(j)vertical bar <= 1 for all (i, j) is an element of A x A. When A = Z(2), the friendly index set of the graph G is defined as {vertical bar e(f)(1) - e(f)(0)vertical bar : the vertex labeling f 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 n not equivalent to 2 (mod 4). (C) 2007 Elsevier B.V. All rights reserved.