Abstract: Let G =( V,E ) be a graph. For a vertex labeling f : V →Z ₂ , it induces an edge labeling f ⁺ : E→Z ₂ , where for each edge v ₁ v ₂ ∈E we have f ⁺ ( v ₁ v ₂ )= f ( v ₁ )+ f ( v ₂ ). For each i∈Z ₂ , we use v _f ( i ) (respectively, e _f ( i )) to denote the number of vertices (respectively, edges) with label i . A vertex labeling f of G is said to be friendly if vertices with different labels differ in size by at most one. The full friendly index set of a graph G , denoted by FFI ( G ), consists of all possible values of e _f (1) −e _f (0), where f ranges over all friendly labelings of G . In this paper, motivated by a problem raised by [6], we study the full friendly index sets of a family of cubic graphs.

Key words: Vertex labeling, Friendly labeling, Embedding labeling graph method, Cubic graph