P2 ▷ H-super antimagic total labeling of comb product of graphs

Abstract

Let L and H be two simple, nontrivial and undirected graphs. Let o be a vertex of H, the comb product between L and H, denoted by L ▷ H, is a graph obtained by taking one copy of L and |V(L)| copies of H and grafting the i th copy of H at the vertex o to the i th vertex of L. By definition of comb product of two graphs, we can say that V(L ▷ H) = {(a, v)|a ∈ V(L), v ∈ V(H)} and (a, v)(b, w) ∈ E(L ▷ H) whenever a = b and vw ∈ E(H), or ab ∈ E(L) and v = w = o. Let G = L ▷ H and P 2 ▷ H ⊆ G, the graph G is said to be an (a, d)- P ▷ H-antimagic total graph if there exists a bijective function f : V(G) ∪ E(G) → {1, 2, . . . , |V(G)| + |E(G)|} such that for all subgraphs isomorphic to P 2 2 ▷ H, the total P ▷ H-weights W( P 2 ▷ H) = ∑ v∈V( P 2 ▷H) f (v) + ∑ f (e) form an arithmetic sequence {a, a +d, a +2d, . . . , a +(n −1)d}, where a and d are positive integers and n is the number of all subgraphs isomorphic to P e∈E( P 2 ▷H) 2 ▷ H. An (a, d)- P ▷ H-antimagic total labeling f is called super if the smallest labels appear in the vertices. In this paper, we study a super (a, d)- P ▷ H-antimagic total labeling of G = L ▷ H when L = C .