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On the vertex irregular total labeling for subdivision of trees
Susilawatia, Baskoro E.T.a, Simanjuntak R.a, Ryan J.b
a Combinatorial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung, 40132, Indonesia
b School of Electrical Engineering and Computing, Faculty of Engineering and Built Environment, The University of Newcastle, Callaghan, 2308, Australia
[vc_row][vc_column][vc_row_inner][vc_column_inner][vc_separator css=”.vc_custom_1624529070653{padding-top: 30px !important;padding-bottom: 30px !important;}”][/vc_column_inner][/vc_row_inner][vc_row_inner layout=”boxed”][vc_column_inner width=”3/4″ css=”.vc_custom_1624695412187{border-right-width: 1px !important;border-right-color: #dddddd !important;border-right-style: solid !important;border-radius: 1px !important;}”][vc_empty_space][megatron_heading title=”Abstract” size=”size-sm” text_align=”text-left”][vc_column_text]© 2018, University of Queensland. All rights reserved.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 (Formula presented). 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) = {e1, 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 ej ∈ 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 (Formula presented) 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).[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Author keywords” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Indexed keywords” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”Funding details” size=”size-sm” text_align=”text-left”][vc_column_text]This research was supported by Program Riset ITB and the PKPI (Sandwich-like) Scholarship from the Ministry of Research, Technology and Higher Education, Indonesia.[/vc_column_text][vc_empty_space][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][vc_empty_space][megatron_heading title=”DOI” size=”size-sm” text_align=”text-left”][vc_column_text][/vc_column_text][/vc_column_inner][vc_column_inner width=”1/4″][vc_column_text]Widget Plumx[/vc_column_text][/vc_column_inner][/vc_row_inner][/vc_column][/vc_row][vc_row][vc_column][vc_separator css=”.vc_custom_1624528584150{padding-top: 25px !important;padding-bottom: 25px !important;}”][/vc_column][/vc_row]