Complete solutions on local antimagic chromatic number of three families of disconnected graphs

Articles in Press

Document Type : Original paper

Authors

1 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, P.R. China.

2 Universiti Teknologi MARA, College of Computing, Informatics & Mathematics, Johor, Malaysia

Abstract

An edge labeling of a graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we study local antimagic labeling of disjoint unions of stars, paths and cycles whose components need not be identical. Consequently, we completely determined the local antimagic chromatic numbers of disjoint union of 2 stars, paths, and 2-regular graphs with at most one odd order component respectively.

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References

[1] S. Arumugam, K. Premalatha, M. Bača, and A. Semaničová-Feňovčíková, Local antimagic vertex coloring of a graph, Graphs Combin. 33 (2017), no. 2, 275–285.
https://doi.org/10.1007/s00373-017-1758-7
[2] M. Bača, A. Semaničová-Feňovčíková, and T.M. Wang, Local antimagic chromatic number for copies of graphs, Mathematics 9 (2021), no. 11, Article ID: 1230.
https://doi.org/10.3390/math9111230
[3] J. Bensmail, M. Senhaji, and K.S. Lyngsie, On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture, Discrete Math. Theor. Comput. Sci. 19 (2017), no. 1, Article ID: #21
https://doi.org/10.23638/DMTCS-19-1-21
[4] T.R. Hagedorn, Magic rectangles revisited, Discrete Math. 207 (1999), no. 1-3, 65–72.
https://doi.org/10.1016/S0012-365X(99)00041-2
[5] T. Harmuth, üeber magische quadrate und ¨ahnliche zahlenfiguren, Arch. Math. Phys 66 (1881), 286–313.
[6] T. Harmuth, üeber magische rechtecke mit ungeraden seitenzahlen, Arch. Math. Phys 66 (1881), 413–447.
[7] J. Haslegrave, Proof of a local antimagic conjecture, Discrete Math. Theor. Comput. Sci. 20 (2018), no. 1, Article ID: #18
https://doi.org/10.23638/DMTCS-20-1-18
[8] G.C. Lau, H.K. Ng, and W.C. Shiu, Affirmative solutions on local antimagic chromatic number, Graphs Combin. 36 (2020), no. 5, 1337–1354.
https://doi.org/10.1007/s00373-020-02197-2
[9] G.C. Lau, W.C. Shiu, and H.K. Ng, On local antimagic chromatic number of cycle-related join graphs, Discuss. Math. Graph Theory 41 (2021), no. 1, 133–152
https://doi.org/10.7151/dmgt.2177
[10] M. Marsidi and I.H. Agustin, The local antimagic on disjoint union of some family graphs, J. Mat. Mantik 5 (2019), no. 2, 69–75.
https://doi.org/10.15642/mantik.2019.5.2.69-75
 
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 16 March 2024

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