On the Total Irregularity Strength of M-Copy Cycles and M-Copy Paths

On Distance Vertex Irregular Total k-Labeling

“…Chartrand et al (1988) first introduced Irregular labeling in 1988, which developed very rapidly since then. Various types of irregular labeling such as VITL (Anholcer et al, 2009;Bača et al, 2007;Indriati et al, 2016;Baskoro et al, 2010), EITL (Anholcer and Palmer, 2012;) Indriati et al, 2013), totally ITL (Indriati et al, 2020;Marzuki et al, 2013), DVIL (Bong et al, 2017;Novindasari et al, 2016;Slamin, 2017;Sugeng et al, 2021;Susanto et al, 2022a;Susanto et al, 2022b), andDVITL (Wijayanti et al, 2021;Wijayanti et al, 2023). Wijayanti et al ( 2021) define the basic concept of DVITL and tdis(G) and also research some necessary and sufficient conditions for the existence of DVITL (Wijayanti et al, 2023) .…”

The total irregularity strength of m copies of the friendship graph

“…Motivated by both labeling, Marzuki, et al [8] introduced a totally irregular total 𝑘-labeling of a graph 𝐺, as a total 𝑘-labeling such that for every two distinct vertices 𝑣 and 𝑥, their weights 𝑤(𝑣) and 𝑤(𝑥) are distinct, and for every two distinct edges 𝑣 1 𝑣 2 and 𝑥 1 𝑥 2 , their weights 𝑤(𝑣 1 𝑣 2 ) and 𝑤(𝑥 1 𝑥 2 ) are distinct. The minimum 𝑘 for which 𝐺 has a totally irregular total 𝑘-labeling is called the total irregularity strength of 𝐺, denoted 𝑡𝑠(𝐺).…”

“…The minimum 𝑘 for which 𝐺 has a totally irregular total 𝑘-labeling is called the total irregularity strength of 𝐺, denoted 𝑡𝑠(𝐺). They [8] provided the lower bound of 𝑡𝑠(𝐺) as follow.…”

On H-Irregularity Strength of Grid Graphs

“…They [2] also decrease the upper bounds of 𝑒ℎ𝑠(𝐺) and 𝑣ℎ𝑠(𝐺) in Theorem A and Theorem B for certain conditions. Later, Ashraf, Baca, Lascsakova, and Semanicova-Fenovcikova in [3] introduced the total 𝐻-irregularity strength of a graph 𝐺, motivated by concepts the total irregularity strength, which is introduced by Marzuki, Salman, and Miller in [12] and 𝐻-covering of a graph 𝐺. A total 𝑘-labeling 𝑓: 𝑉 ∪ 𝐸 → {1, 2, … , 𝑘} of 𝐺 is called a totally irregular total 𝑘-labeling if for any pair of vertices 𝑥 and 𝑦, their weights 𝑤(𝑥) and 𝑤(𝑦) are distinct and for any pair of edges 𝑥 1 𝑥 2 and 𝑦 1 𝑦 2 , their weights 𝑤(𝑥 1 𝑥 2 ) and 𝑤(𝑦 1 𝑦 2 ) are distinct.…”

“…The minimum 𝑘 for which a graph 𝐺 has totally irregular total labeling, is called total irregularity strength of 𝐺, denoted by 𝑡𝑠(𝐺). They [12] gave the lower bound of 𝑡𝑠(𝐺) and provided the exact value of 𝑡𝑠(𝐺) for cycles and paths. For fan, wheel, triangular book, and friendship graphs, Tilukay, Salman, and Persulessy [14] gave the exact values of their 𝑡𝑠 which is equal to the lower bound, as well as Tilukay, Tomasouw, Rumlawang, and Salman gave in [15] for complete and complete bipartite graphs,while Ramdani and Salman [13] gave the exact values of 𝑡𝑠 of some Cartesian products graphs.…”