On friendly index sets of k-galaxies

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.