- Title
- On the vertex irregular total labeling for subdivision of trees
- Creator
- Susilawati,; Baskoro, Edy Tri; Simanjuntak, Rinovia; Ryan, Joe
- Relation
- Australasian Journal of Combinatorics Vol. 71, Issue 2, p. 293-302
- Relation
- https://ajc.maths.uq.edu.au/pdf/71/ajc_v71_p293.pdf
- Publisher
- Centre for Discrete Mathematics & Computing
- Resource Type
- journal article
- Date
- 2018
- Description
- Let G = (V, E) be a simple, connected and undirected graph with nonempty vertex set V (G) and edge set E(G). We define a labeling φ : V ∪ E → {1, 2, 3, . . . , k} to be a vertex irregular total k-labeling of G if for every two different vertices x and y of G, their weights w(x) and w(y) are distinct, where the weight w(x) of a vertex x ∈ V is w(x) = φ(x) + P xy∈E(G) φ(xy). The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, denoted by tvs(G). The subdivision graph S(G) of a graph G is the graph obtained from G by replacing each edge e = uv with the path (u, re, v) of length 2, where re is a new vertex (called a subdivision vertex) corresponding to the edge e. Let T be a tree. Let E(T) = E1 ∪ E2 be the set of edges in T where E1(T) = {e1, e2, . . . , en1 } and E2(T) = {e 1, e2, . . . , en2 } are the sets of pendant edges and interior edges, respectively. Let S(T; ri; sj ) be the subdivision tree obtained from T by replacing each edge ei ∈ E1 with a path of length ri + 1 and each edge e j ∈ E2 with a path of length sj + 1, for i ∈ [1, n1] and j ∈ [1, n2]. In 2010, Nurdin et al. conjectured that tvs(T) = max{t1, t2, t3}, where ti = d(1 +Pi j=1 nj )/(i+ 1)e and ni is the number of vertices of degree i ∈ [1, 3]. In this paper, we show that the total vertex irregularity strength of S(T; ri; sj ) is equal to t2, where the value of t2 is calculated for S(T; ri; sj ).
- Subject
- vertex irregular; weight; graph; edges
- Identifier
- http://hdl.handle.net/1959.13/1448167
- Identifier
- uon:43331
- Identifier
- ISSN:1034-4942
- Language
- eng
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