Formula omitted]-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 ith copy of H at the vertex o to the ith 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 P2▹H⊆G, the graph G is said to be an (a,d)-P2▹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 P2▹H, the total P2▹H-weights W(P2▹H)=∑v∈V(P2▹H)f(v)+∑e∈E(P2▹H)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 P2▹H. An (a,d)-P2▹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)-P2▹H-antimagic total labeling of G=L▹H when L=Cn.