The antimagicness of super (a, d) − P2⊵̇H total covering on total comb graphs

Abstract

A graph can be constructed in several ways. One of them is by operating two or more graphs. The resulting graphs will be a new graph which has certain characteristics. One of the latest graph operations is total comb of two graphs. Let L, H be a finite collection of nontrivial, simple and undirected graphs. The total comb product is a graph obtained by taking one copy of L and |V(L)| + |E(L)| copies of H and grafting the i-th copy of H at the vertex o and edge uv to the i-th vertex and edge of L. 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) = Σv∈V(P2⊵̇H) f (v) + Σe∈E(P2⊵̇H) f (e) form an arithmetic sequence. An (a, d)-P2⊵̇H-antimagic total covering f is called super when the smallest labels appear in the vertices. By using partition technique has been proven that the graph G = L⊵̇H admits a super (a, d)-P2⊵̇H antimagic total labeling with different value d = d∗ + d∗(dv1 + de1) + dv2 + de2 + 1.