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On the total irregularity strength of cycles and paths
Marzuki C.C.a, Salman A.N.M.a, Miller M.b,c,d
a Combinatorial Mathematics Research Group, Institut Teknologi Bandung, Indonesia
b Department of Informatics, King’s College London, United Kingdom
c Department of Mathematics, University of West Bohemia, Czech Republic
d School of Electrical Engineering and Computer Science, University of Newcastle, 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]The vertex irregular total labeling and the edge irregular total labeling were introduced by Bača et al. in [5]. Combining both of these notions, in this paper, we introduce a new irregular total labeling, called ‘totally irregular total labeling’ which is required to be both vertex and edge irregular. Let G = (V, E) be a graph. A function f : V∪E → {1, 2, …, k} of a graph G is a totally irregular total k-labeling if for any two different vertices x and y of G, their weights wt(x) and wt(y) are distinct and for any two different edges x1x2 and y1y2 of G, their weights wt(x1x2) and wt(y1y2) are distinct, where the weight wt(x) of a vertex x is the sum of the label of x and the labels of all edges incident with x, and the weight wt(x1x2) of an edge x1x2 is the sum of the label of edge x1x2 and the labels of vertices x1 and x2. The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G). In this paper, we provide an upper bound and a lower bound of the total irregularity strength of a graph. Besides that, we determine the total irregularity strength of cycles and paths. © 2013 Pushpa Publishing House, Allahabad, India.[/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]Cycle,Path,Total irregularity strength,Totally irregular total k-labeling[/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][/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]