3-vertex friendly index set of graphs

Graph labeling is an assignment of integers to the vertices or the edges, or both, subject to certain conditions. In literature we find several labelings such as graceful, harmonious, binary, friendly, cordial, ternary and many more. A friendly labeling is a binary mapping such that |v_f (1) − v_f (0)|≤ 1 where v_f (1) and v_f (0) represents number of vertices labeled by 1 and 0 respectively. For each edge uv assign the label |f(u) − f(v)|, then the function f is cordial labeling of G if |v_f (1) − v_f (0)|≤ 1 and |e_f^∗ (1) − e_f^∗ (0)|≤ 1, where e_f^∗ (1) and e_f^∗ (0) are the number of edges labeled 1 and 0 respectively. A friendly index set of a graph is {|e_f^∗ (1) − e_f^∗(0)|: f^∗ runs over all friendly labeling f of G} and it is denoted by F I(G). A mapping f: V (G) → {0, 1, 2} is called ternary vertex labeling and f(v) represents the vertex label for v. In this article, we extend the concept of ternary vertex labeling to 3-vertex friendly labeling and define 3-vertex friendly index set of graphs. The set F I_3v (G) = {|e_f^∗ (i) − e_f^∗ (j)|: f^∗ runs over all 3 − vertex friendly labeling f for all i, j ∈ {0, 1, 2}} is referred as 3-vertex friendly index set. In order to achieve F I_3v (G), number of vertices are partitioned into {V₀, V₁, V₂ } such that ||V_i |−|V_j ||≤ 1 for all i, j = 0, 1, 2 with i ≠ j and label the edge uv by |f(u) −f(v)| where f(u), f(v) ∈ {0, 1, 2}. In this paper, we study the 3-vertex friendly index sets of some standard graphs such as complete graph K_n, path P_n, wheel graph W_n, complete bipartite graph K_m,n and cycle with parallel chords P C_n .