On super H−antimagicness of an edge comb product of graphs with subgraph as a terminal of its amalgamation

Abstract

All graphs in this paper are simple, finite, and undirected graph. Let r be a edges of H. The edge comb product between L and H, denoted by L⊵H, is a graph obtained by taking one copy of L and |E(L)| copies of H and grafting the i-th copy of H at the edges r to the i-th edges of L, we call such a graph as an edge comb product of graph with subgraph as a terminal of its amalgamation, denoted by G = K⊵Amal(H, L ⊂ H, n). The graph G is said to admits an (a, d)-H-antimagic total labeling if there exist a bijection f : V(G) ∪ E(G) → {1, 2, …, |V (G)| + |E(G)|} such that for all subgraphs isomorphic to H, the total H-weights W (H) = ∑v∈V(H) f(v) + ∑e∈E(H) f(e) form an arithmetic sequence {a, a + d, a + 2d, …, a + (t − 1)d}, where a and d are positive integers and t is the number of all subgraphs isomorphic to H. An (a, d)-H-antimagic total labeling f is called super if the smallest labels appear in the vertices. In this paper, we will study the super H−antimagicness of disjoint union of edge comb product of graphs with subgraph as a terminal of its amalgamation.