International Journal of Advanced Engineering Research and Science (IJAERS) https://dx.doi.org/10.22161/ijaers.4.5.32
[Vol-4, Issue-5, May- 2017] ISSN: 2349-6495(P) | 2456-1908(O)
Super complete-antimagicness of Amalgamation of any Graph R.M Prihandini1,3, Ika Hesti Agustin1,4, Dafik1,2, Ridho Alfarisi1,3 1
CGANT-University of Jember, Jember, Indonesia Department of Mathematics Education, University of Jember, Jember, Indonesia 3 Department of Elementary School Teacher Education, University of Jember, Jember, Indonesia 4 Department of Mathematics, University of Jember, Jember, Indonesia 2
Abstract—Let đ??ťđ?‘– be a finite collection of simple, nontrivial and undirected graphs and let each đ??ťđ?‘– have a fixed vertex đ?‘Łđ?‘— called a terminal. The amalgamation đ??ťđ?‘– as đ?‘Łđ?‘— as a terminal is formed by taking all the đ??ťđ?‘– 's and identifying their terminal. When đ??ťđ?‘– are all isomorphic graphs, for any positif integer đ?‘›we denote such amalgamation by đ??ş = đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›), where đ?‘› denotes the number of copies of đ??ť. The graph đ??ş is said to be an (đ?‘Ž, đ?‘‘) − đ??ť-antimagic total graph if there exist a bijective function đ?‘“: đ?‘‰(đ??ş) âˆŞ đ??¸(đ??ş) → {1, 2, ‌ , |đ?‘‰ (đ??ş)| + |đ??¸(đ??ş)|} such that for all subgraphs isomorphic to đ??ť, the total đ??ť −weights đ?‘Š(đ??ť) = ∑đ?‘Łâˆˆ đ?‘‰(đ??ť) đ?‘“(đ?‘Ł) + ∑đ?‘’∈ đ??¸(đ??ť) đ?‘“(đ?‘’) form an arithmetic sequence {đ?‘Ž, đ?‘Ž + đ?‘‘, đ?‘Ž + 2đ?‘‘, . . . , đ?‘Ž + (đ?‘› − 1)đ?‘‘}, wheređ?‘Ž and đ?‘‘ are positive integers and đ?‘› is the number of all subgraphs isomorphic to đ??ť. An (đ?‘Ž, đ?‘‘) − đ??ť-antimagic total labeling đ?‘“is called super if the smallest labels appear in the vertices. In this paper, we study a super (đ?‘Ž, đ?‘‘) − đ??ť antimagic total labeling of đ??ş = đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›) and its disjoint union. Keywords—Super đ??ť-antimagic total graph, Amalgamation of graph, arithmetic sequence. I. INTRODUCTION A graph đ??ş is said to be an (đ?‘Ž, đ?‘‘) − đ??ť-antimagic total graph if there exist a bijective function đ?‘“: đ?‘‰(đ??ş) âˆŞ đ??¸(đ??ş) → {1,2, ‌ , |đ?‘‰ (đ??ş)| + |đ??¸(đ??ş)|} such that for all subgraphs of đ??şisomorphic to đ??ť, the total đ??ť-weights đ?‘¤(đ??ť) = ÎŁđ?‘Łâˆˆđ?‘‰(đ??ť) đ?‘“(đ?‘Ł) + ÎŁđ?‘’∈ đ??¸(đ??ť) đ?‘“(đ?‘’) form an arithmetic sequence {đ?‘Ž, đ?‘Ž + đ?‘‘, đ?‘Ž + 2đ?‘‘, . . . , đ?‘Ž + (đ?‘› − 1)đ?‘‘}, where đ?‘Ž and đ?‘‘are positive integers and đ?‘› is the number of all subgraphs of đ??ş isomorphic tođ??ť. If such a function exist then đ?‘“ is called an(đ?‘Ž, đ?‘‘) − đ??ť-antimagic total labeling of đ??ş. An(đ?‘Ž, đ?‘‘) − đ??ť -antimagic total labeling đ?‘“ is called super if đ?‘“: đ?‘‰(đ??ş) → {1, 2, ‌ , |đ?‘‰ (đ??ş)|}. There many articles have been published in many journals, some ofthem can be cited in [2, 3, 7, 8] and [9, 10, 11, 12, 13]. For connected graph, Inayah et al. in [7] proved that, for đ??ť is a non-trivial connected graph and đ?‘˜ ≼ 2 is an integer, đ?‘ â„Žđ?‘Žđ?‘?đ?‘˜(đ??ť, đ?‘Ł, đ?‘˜) which contains exactly đ?‘˜ subgraphs isomorphic to đ??ť is đ??ť-super antimagic. They www.ijaers.com
only covered a connected version of shackle of graph when a vertex as a connector, and their paper did not cover all feasible đ?‘‘. Our paper attempt to solve a super (đ?‘Ž, đ?‘‘) − đ??ťantimagic total labeling of đ??ş = đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›) and its disjoint union when đ??ť is a complete graph for feasible đ?‘‘. To show those existence, we will use a special technique, namely an integer set partition technique. We consider the đ?‘› partition đ?‘ƒđ?‘š,đ?‘‘ (đ?‘–, đ?‘—) of the set {1, 2, ‌ , đ?‘šđ?‘›} into đ?‘› columns with đ?‘› ≼ 2, đ?‘š-rows such that the difference between the sum of the numbers in the (đ?‘— + 1)th đ?‘š-rows and the sum of the numbers in the đ?‘—th đ?‘š-rows is always equal to the constant đ?‘‘,where đ?‘— = 1, 2, ‌ , đ?‘› − 1. The đ?‘›,đ?‘ partitionđ?‘ƒđ?‘š,đ?‘‘ (đ?‘–, đ?‘—, đ?‘˜) of the set {1, 2, ‌ , đ?‘šđ?‘›đ?‘ }into đ?‘›đ?‘ columns with đ?‘›, đ?‘ ≼ 2, đ?‘š-rows such that the difference between the sum of the numbers in the (đ?‘˜ + 1)th đ?‘š-rows and the sum of the numbers in the đ?‘˜th đ?‘š-rows is always equal to the constant đ?‘‘ for đ?‘— = 1, 2, ‌ , đ?‘›, where đ?‘˜ = 1, 2, ‌ , đ?‘˜ − 1. Thus these sums form an arithmetic sequence with the difference đ?‘‘. We need to establish some đ?‘›,đ?‘ đ?‘› lemmas related to the partition đ?‘ƒđ?‘š,đ?‘‘ (đ?‘–, đ?‘—)and đ?‘ƒđ?‘š,đ?‘‘ (đ?‘–, đ?‘—, đ?‘˜). These lemmas are useful to develop the super (đ?‘Ž, đ?‘‘) − đ??ť antimagic total labeling of đ??ş = đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›)and đ??ş = đ?‘ đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›). II. SOME USEFUL LEMMAS Let G be an amalgamation of any graph H, denoted by đ??ş = đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›). The graph đ??ş is a connected graph with |đ?‘‰(đ??ş)| = đ?‘?đ??ş , |đ??¸(đ??ş)| = đ?‘žđ??ş , |đ?‘‰(đ??ť)| = đ?‘?đ??ť , and đ??¸(đ??ť)| = đ?‘žđ??ť . The vertex set and edge set of the graph đ??ş = đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›) can be split into following sets: đ?‘‰(đ??ş) = {đ??´} âˆŞ {đ?‘Ľđ?‘–đ?‘— ; 1 ≤ đ?‘– ≤ đ?‘?đ??ť − 1 , 1 ≤ đ?‘— ≤ đ?‘› } and đ??¸(đ??ş) = {đ?‘’đ?‘™đ?‘— ; 1 ≤ đ?‘™ ≤ đ?‘žđ??ť , 1 ≤ đ?‘— ≤ đ?‘› }. Let đ?‘›, đ?‘š be positive integers with đ?‘› ≼ 2 and đ?‘š ≼ 3. Thus |đ?‘‰(đ??ş)| = đ?‘?đ??ş = đ?‘›(đ?‘?đ??ť − 1) + 1and |đ??¸(đ??ş)| = đ?‘žđ??ş = đ?‘›đ?‘žđ??ť . Furthermore, let đ??ş be a disjoint union of amalgamation of graph đ??ť, denoted by đ??ş = đ?‘ đ??´đ?‘šđ?‘Žđ?‘™(đ??ť, đ?‘Ł, đ?‘›) and đ?‘ bean odd positive integer. The graph G is a disconnected graph with |đ?‘‰(đ??ş)| = đ?‘?đ??ş , |đ??¸(đ??ş)| = đ?‘žđ??ş , |đ?‘‰(đ??ť)| = đ?‘?đ??ť , and |đ??¸(đ??ť)| = đ?‘žđ??ť . The vertex set and edge set of the graph đ??ş = Page | 202