On friendly index sets of k-galaxies

Abstract

Let G = ( V , E ) be a graph. A vertex labeling f : V → Z 2 induces an edge labeling f * : E → Z 2 defined by f * ( x y ) = f ( x ) + f ( y ) , for each edge x y ∈ E . For i ∈ Z 2 , let v f ( i ) = ∣{ v ∈ V : f ( v ) = i }∣ and e f ( i ) = ∣{ e ∈ E : f * ( e ) = i }∣ . We say that f is f r i e n d l y if ∣ v f (1) − v f (0)∣ ≤ 1 . The friendly index set of G , denoted by FI ( G ) , is defined as FI( G ) = ∣ e f (1) − e f (0)∣ : vertex labeling f is friendly . A k -galaxy is a disjoint union of k stars. In this paper, we establish the friendly index sets for various classes of k -galaxies.