The Local Antimagic On Disjoint Union of Some Family Graphs

  • Marsidi Marsidi IKIP PGRI JEMBER
  • Ika Hesti Agustin Universitas Jember
Keywords: local antimagic labelling; local antimagic chromatic number, disjoint union graphs

Abstract

A graph  in this paper is nontrivial, finite, connected, simple, and undirected. Graph  consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of  is not equal with the weight of , where the weight of  (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where  is the colour on the vertex . The vertex local antimagic chromatic number  is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Downloads

Download data is not yet available.

References

J. L. Gross, J. Yellen, and P. Zhang, Handbook of graph Theory Second Edition. CRC Press Taylor and Francis Group, 2014

G. Chartrand, L. Lesniak, and P. Zhang, Graphs & Digraphs 6th Edition. CRC Press: Taylor & Francis Group, 2016

D. Zaenab, D. Adyanti, A. Fanani, and N. Ulinnuha, “Aplikasi Graph Coloring pada Penjadwalan Perkuliahan di Fakultas Sains dan Teknologi UIN Sunan Ampel Surabaya”, mantik, vol. 2, no. 1, pp. 30-39, October 2016.

J. L. Gross, J. Yellen, and P. Zhang, Handbook of Graph Theory Second Edition. CRC Press Taylor and Francis Group, 2014

S. Arumugam S, K. Premalatha, M. Baca and A. Semanicova-Fenovcikova, "local antimagic vertex colouring of a graph", Graphs and Combinatorics Volume 33, Issue 2, pp. 275-285, 2017

I. H. Agustin, Dafik, M. Hasan, R. Alfarisi R, and R. M. Prihandini, "local edge antimagic colouring of graphs". in Far East Journal of Mathematical Sciences, 102(9):1925-1941, 2017

I. H. Agustin, S. Dafik, R. Alfarisi, E.Y. Kurniawati, "The construction of super local edge antimagic total coloring by using an EAVL Technique accepted", 2017

I. H. Agustin, S. Dafik, E. R. Ermita, R. Alfarisi, "On the total local edge super antimagicness of special graph and graph with pendant edge accepted, 2017

E. Y. Kurniawati, I. H. Agustin, Dafik, and R. Alfarisi. Super local edge antimagic total colouring of {P}_{n}vartriangleright H, Journal of Physics: Conference Series, Volume 1008, Issue 1, 2018

E. Y. Kurniawati, I. H. Agustin, Dafik, R. Alfarisi, and Marsidi, "On the local edge antimagic total chromatic number of amalgamation of graphs", AIP Conference Proceedings, 2018

Yung-Ling Lai and G. J. Chang, "On the profile of the corona of two graphs, Information Processing Letters, Vol. 89, Issue 6, pp. 287-292, 2004.

CROSSMARK
Published
2019-10-27
DIMENSIONS
How to Cite
MarsidiM., & AgustinI. H. (2019). The Local Antimagic On Disjoint Union of Some Family Graphs. Jurnal Matematika MANTIK, 5(2), 69-75. https://doi.org/10.15642/mantik.2019.5.2.69-75