Local distance antimagic cromatic number of join product of graphs with cycles or paths

Year 2024, , 788 - 802, 27.06.2024

Wai-chee Shiu , Gee-choon Lau , Nalliah M

https://doi.org/10.15672/hujms.1266085

Abstract

Let $G$ be a graph of order $p$ without isolated vertices. A bijection $f: V \to \{1,2,3,\dots,p\}$ is called a local distance antimagic labeling, if $w_f(u)\ne w_f(v)$ for every edge $uv$ of $G$, where $w_f(u)=\sum_{x\epsilon N(u)} {f(x)}$. The local distance antimagic chromatic number $\chi_{lda}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local distance antimagic labelings of $G$. In this paper, we determined the local distance antimagic chromatic number of some cycles, paths, disjoint union of 3-paths. We also determined the local distance antimagic chromatic number of join products of some graphs with cycles or paths.

Keywords

local distance antimagic chromatic number, join product, cycle, path, null graph

References

• [1] S. Arumugam, D. Froncek, and N. Kamatchi, Distance magic graphs–A survey, J. Indones. Math. Soc. Special Edition, 1126, 2011.
• [2] S. Arumugam and N. Kamatchi, On $(a, d)$-distance antimagic graphs, Australas. J. Combin. 54, 279–287, 2012.
• [3] S. Arumugam, K. Premalatha, M. Bača and A. Semaničová-Fecňovčíková, Local antimagic vertex coloring of a graph, Graphs Combin. 33, 275–285, 2017.
• [4] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, New York, MacMillan, 1976.
• [5] 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 (1), 2017.
• [6] T. Divya and S. Devi Yamini, Local distance antimagic vertex coloring of graphs, https://arxiv.org/abs/2106.01833v1, 2021.
• [7] J.A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin.1 (Dynamic Surveys),DS6, 2021.
• [8] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, INC., Boston, 1994.
• [9] J. Haslegrave, Proof of a local antimagic conjecture, Discrete Math. Theor. Comput. Sci. 20 (1), 2018.
• [10] N. Kamatchi and S. Arumugam, Distance antimagic graphs, J. Combin. Math. Combin. Comput. 84, 6167, 2013.
• [11] G.C. Lau, H.K. Ng and W.C. Shiu, Affirmative solutions on local antimagic chromatic number, Graphs Combin. 36, 1337–1354, 2020.
• [12] G.C. Lau, W.C. Shiu and H.K. Ng, On local antimagic chromatic number of cyclerelated join graphs, Discuss. Math. Graph Theory, 4 (1), 133–152, 2021.
• [13] M. Nalliah, Antimagic labeling of Graphs and digraphs, Ph.D Thesis, Kalasalingam University, Tamil Nadu, India, 2013.
• [14] S. Shaebani, On local antimagic chromatic number of graphs, J. Algebr. Syst. 7 (2), 245–256, 2020.
• [15] R. Shankar and M. Nalliah, Local vertex antimagic chromatic number of some wheel related graphs, Proyecciones J. Math. 41 (1), 319–334, 2022.
• [16] V. Priyadharshini and M. Nalliah, Local distance antimagic chromatic number for the union of complete bipartite graphs, Tamkang J. Math. online, https://journals.math.tku.edu.tw/index.php/TKJM/article/view/4804, 2023.