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Super (a, d)-H-antimagic total labelings for shackles of a connected graph H
Inayah N.a,b, Simanjuntak R.a, Salman A.N.M.a, Syuhada K.I.A.a
a Combinatorial Mathematics Research Group, Institut Teknologi Bandung, Indonesia
b Program Studi Matematika, FST, Universitas Islam Negeri (UIN) Syarif Hidayatullah Jakarta, Indonesia
[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]A simple graph G = (V (G), E(G)) admits an H-covering, if every edge in E(G) belongs to at least one subgraph of G isomorphic to a given graph H. An (a, d)-H-antimagic total labeling of G admitting an H-covering is a bijective function ξ: V (G) ∪ E(G) → {1, 2,…, {pipe}V(G){pipe} + {pipe}E(G){pipe}} such that for all subgraphs H′ isomorphic to H, the H-weights w(H′) = ΣvεV(H′)ξ(v) + ΣeεE(H′)ξ(e) constitute an arithmetic progression a, a + d, a + 2d,…, a + (k – 1)d where a and d are positive integers and k is the number of subgraphs of G isomorphic to H. Such a labeling is called super if the smallest possible labels appear on the vertices. This paper is devoted to studying super (a, d)-H-antimagic total labelings for some shackles of a connected graph H. A shackle of G1,G2,…, Gk, denoted by shack(G1,G2,…, Gk), is a graph constructed from nontrivial connected and ordered graphs G1,G2,…,Gk such that for every 1 ≤ i, j ≤ k with {pipe}i – j{pipe} ≥ 2, Gi and Gj have no common vertex, and for every 1 ≤ i ≤ k – 1, Gi and Gi+1 share exactly one common vertex, called a linkage vertex, where the k – 1 linkage vertices are all distinct. In the case when all Gi’s are isomorphic to a connected graph H, we call the resulting graph a shackle of H, denoted by shack(H, k). A technique of partitioning a multiset is introduced and an upper bound for d is obtained.[/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][/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]