On friendly index sets of 2-regular graphs

Abstract

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 ∗ ( x y ) = 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 = Z 2 , the friendly index set of the graph G is defined as { | e f ( 1 ) − e f ( 0 ) | : 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 ≢ 2 (mod 4).