$Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union$

Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union

Let M = (V(M), E(M)) be a simple graph with finite vertices and edges. An N-covering of M is a family {N1, N2, ... , Nα} of subgraphs of M isomorphic to N such that every edge in E(M) belongs to Nl...

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Main Authors:	Yi Hu, Muhammad Awais Umar, Mustafa Habib, Ce Shi, Ghulam Rasool, Zijiang Zhu
Format:	Article
Language:	English
Published:	IEEE 2019-01-01
Series:	IEEE Access
Subjects:	Block cut-vertex graph <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-path disjoint union of <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-path super (<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">c</italic>, <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">d</italic>)-<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">cₙ</italic>-antimagic
Online Access:	https://ieeexplore.ieee.org/document/8880653/

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record_format	Article
spelling	doaj-69a61b26a82b4656a5eb2c161d855c9f2021-03-30T00:51:18ZengIEEEIEEE Access2169-35362019-01-01715529915530310.1109/ACCESS.2019.29490438880653Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint UnionYi Hu0Muhammad Awais Umar1Mustafa Habib2https://orcid.org/0000-0002-2797-1865Ce Shi3Ghulam Rasool4https://orcid.org/0000-0001-8681-0343Zijiang Zhu5School of Information Science and Technology, South China Business College of Guangdong University of Foreign Studies, Guangzhou, ChinaGovernment Degree College (B), Sharaqpur Sharif, PakistanDepartment of Mathematics, University of Engineering and Technology Lahore, Lahore, PakistanSchool of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai, ChinaDepartment of Mathematics, National College of Business Administration & Economics (NCBA&E), DHA Campus, Lahore, PakistanSchool of Information Science and Technology, South China Business College of Guangdong University of Foreign Studies, Guangzhou, ChinaLet M = (V(M), E(M)) be a simple graph with finite vertices and edges. An N-covering of M is a family {N<sub>1</sub>, N<sub>2</sub>, ... , N<sub>α</sub>} of subgraphs of M isomorphic to N such that every edge in E(M) belongs to N<sub>l</sub>, for some l, l ∈ {1, 2, ... , α}. Such a graph is a (c, d)-N-antimagic if ∃ a bijection ψ : V<sub>M</sub> ∪E<sub>M</sub> → {1, 2, ... , \|V<sub>M</sub>\|+\|E<sub>M</sub>\|} such that for all N<sub>l</sub> ≅ N, {wt<sub>ψ</sub>(N<sub>l</sub>)} = {c, c+d, ... , c+(α-1)d}. For ψ(V(M)) = {1, 2, 3, ... , \|V(M)\|}, the labeling ψ would be super (c, d)-N-antimagic and for d = 0 it would be N-supermagic. In this manuscript, we investigated that mC<sub>n</sub>-path has super (c, d)-C<sub>n</sub>-antimagic labeling for differences d ∈ {0, 1, ... , 5} and extend the result for C<sub>n</sub>-supermagic labeling of disjoint union of isomorphic copies of mC<sub>n</sub>-path.https://ieeexplore.ieee.org/document/8880653/Block cut-vertex graph<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-pathdisjoint union of <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-pathsuper (<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">c</italic>, <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">d</italic>)-<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">cₙ</italic>-antimagic
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Yi Hu Muhammad Awais Umar Mustafa Habib Ce Shi Ghulam Rasool Zijiang Zhu
spellingShingle	Yi Hu Muhammad Awais Umar Mustafa Habib Ce Shi Ghulam Rasool Zijiang Zhu Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union IEEE Access Block cut-vertex graph <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-path disjoint union of <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-path super (<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">c</italic>, <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">d</italic>)-<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">cₙ</italic>-antimagic
author_facet	Yi Hu Muhammad Awais Umar Mustafa Habib Ce Shi Ghulam Rasool Zijiang Zhu
author_sort	Yi Hu
title	Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union
title_short	Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union
title_full	Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union
title_fullStr	Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union
title_full_unstemmed	Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union
title_sort	antimagicness of <inline-formula> <tex-math notation="latex">$mc_n$ </tex-math></inline-formula>-path and its disjoint union
publisher	IEEE
series	IEEE Access
issn	2169-3536
publishDate	2019-01-01
description	Let M = (V(M), E(M)) be a simple graph with finite vertices and edges. An N-covering of M is a family {N<sub>1</sub>, N<sub>2</sub>, ... , N<sub>α</sub>} of subgraphs of M isomorphic to N such that every edge in E(M) belongs to N<sub>l</sub>, for some l, l ∈ {1, 2, ... , α}. Such a graph is a (c, d)-N-antimagic if ∃ a bijection ψ : V<sub>M</sub> ∪E<sub>M</sub> → {1, 2, ... , \|V<sub>M</sub>\|+\|E<sub>M</sub>\|} such that for all N<sub>l</sub> ≅ N, {wt<sub>ψ</sub>(N<sub>l</sub>)} = {c, c+d, ... , c+(α-1)d}. For ψ(V(M)) = {1, 2, 3, ... , \|V(M)\|}, the labeling ψ would be super (c, d)-N-antimagic and for d = 0 it would be N-supermagic. In this manuscript, we investigated that mC<sub>n</sub>-path has super (c, d)-C<sub>n</sub>-antimagic labeling for differences d ∈ {0, 1, ... , 5} and extend the result for C<sub>n</sub>-supermagic labeling of disjoint union of isomorphic copies of mC<sub>n</sub>-path.
topic	Block cut-vertex graph <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-path disjoint union of <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">mCₙ</italic>-path super (<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">c</italic>, <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">d</italic>)-<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">cₙ</italic>-antimagic
url	https://ieeexplore.ieee.org/document/8880653/
work_keys_str_mv	AT yihu antimagicnessofinlineformulatexmathnotationlatexmcntexmathinlineformulapathanditsdisjointunion AT muhammadawaisumar antimagicnessofinlineformulatexmathnotationlatexmcntexmathinlineformulapathanditsdisjointunion AT mustafahabib antimagicnessofinlineformulatexmathnotationlatexmcntexmathinlineformulapathanditsdisjointunion AT ceshi antimagicnessofinlineformulatexmathnotationlatexmcntexmathinlineformulapathanditsdisjointunion AT ghulamrasool antimagicnessofinlineformulatexmathnotationlatexmcntexmathinlineformulapathanditsdisjointunion AT zijiangzhu antimagicnessofinlineformulatexmathnotationlatexmcntexmathinlineformulapathanditsdisjointunion
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Antimagicness of <inline-formula> <tex-math notation="LaTeX">$mC_n$ </tex-math></inline-formula>-Path and Its Disjoint Union

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