P2▹H-super antimagic total labeling of comb product of graphs. Issue 2 (1st August 2019)

• Record Type: Journal Article
• Authors: Agustin, Ika Hesti; Prihandini, R.M.; Dafik,
• Is Part Of: AKCE International Journal of Graphs and Combinatorics. Volume 16:Issue 2(2019)
• Page Start: 163
• Page End: 171
• Publication Date: 2019-08-01
• Subjects: Super H-antimagic total labeling -- Comb product -- Cycle graph
• DOI: 10.1016/j.akcej.2018.01.008
• Languages: English
• ISSNs: 0972-8600

Abstract

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 v w ∈ E ( H ), or a b ∈ 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 2 ▹ 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 ▹ H, the total P 2 ▹ H -weights W ( P 2 ▹ H ) = ∑ v ∈ V ( P 2 ▹ H ) f ( v ) + ∑ e ∈ E ( P 2 ▹ H ) f ( e ) form an arithmetic sequence { a, a + d, a + 2 d, …, a + ( n − 1 ) d }, where a and d are positive integers and n is the number of all subgraphs isomorphic to P 2 ▹ H . An ( a, d ) - P 2 ▹ 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 2 ▹ H -antimagic total labeling of G = L ▹ H when L = C n .