Totally irregular total labeling of some caterpillar graphs

Abstract

Assume that G(V, E) is a graph with V and E as its vertex and edge sets, respectively. We have G is simple, connected, and undirected. Given a function λ from a union of V and E into a set of k-integers from 1 until k. We call the function λ as a totally irregular total k-labeling if the set of weights of vertices and edges consists of different numbers. For any u ∈ V , we have a weight wt(u) = λ(u) + uy∈E λ(uy). Also, it is defined a weight wt(e) = λ(u) + λ(uv) + λ(v) for each e = uv ∈ E. A minimum k used in k-total labeling λ is named as a total irregularity strength of G, symbolized by ts(G). We discuss results on ts of some caterpillar graphs in this paper. The results are ts(S p,2,2,q ) = p+q−1 2 for p, q greater than or equal to 3, while ts(S p,2,2,2,p ) = 2p−1 2 , p ≥ 4.

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) .…”

A survey on enhanced power graphs of finite groups

“…The investigation of graph labelings forms a large and important research direction. Many valuable classes of graph labelings have been investigated in the literature, including cordial labelings [10], distance labelings [66], face-antimagic labelings [9], graceful labelings [57], integer additive labelings [71], irregular total labelings [46], supermagic labelings [64] and vertex irregular reflexive labelings [8,75]. The following general problem is an interesting suggestion for future work.…”

“…Therefore, this direction can also be considered as a part of the broader field of research -the investigation of graph labelings. For more information, refer to examples of publications devoted to graph labelings ( [10,46,57,66,71]).…”

Modular irregularity strength on some flower graphs

“…This graph invariant has attracted much attention see [1,2,6,8,9,11,12]. People alsi study the variation on irregular labeling, as an example total irregular labeling where the label is definied for vertices and edges, see [10,13] as examples.…”