Total vertex irregularity strength of comb product of two cycles

Abstract

Let G = (V (G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V (G) ∪ E(G) → {1,2...,k}. The vertex weight v under the labeling f is denoted by Wf(v) and defined by Wf(v) = f(v) + Σuv∈E(G)f(uv). A total k-labeling of G is called vertex irregular if there are no two vertices with the same weight. The total vertex irregularity strength of G, denoted by tvs(G), is the minimum k such that G has a vertex irregular total k-labeling. This labeling was introduced by Bača, Jendrol', Miller, and Ryan in 2007. Let G and H be two connected graphs. Let o be a vertex of H . The comb product between G and H, in the vertex o, denoted by G⊳o H, is a graph obtained by taking one copy of G and |V (G)| copies of H and grafting the i-th copy of H at the vertex o to the i-th vertex of G. In this paper, we determine the total vertex irregularity strength of comb product of Cn and Cm where m ∈ {1,2}.

Nilai Total Ketakteraturan Titik dari Honeycomb Network

“…Kemudian, pada tahun 2018 Ramdani dkk. [20] mendapatkan nilai total ketakteraturan titik untuk graf hasil kali comb dari dua graf lingkaran. Sedangkan Jeyanthy dkk.…”

On the total vertex irregularity strength of comb product of two cycles and two stars

“…for connected graph G having n i vertices of degree i(i = δ, δ + 1, δ + 2, · · · , ∆), where δ and ∆ are the minimum and the maximum degree of G, respectively. In [8], Ramdani and Ramdhani obtained the exact value of the total vertex irregularity strength of comb product between cycles C n and C 4 , as follows.…”