Local distance antimagic chromatic number for the union of star and double star graphs

V.  Priyadharshini; M.  Nalliah

doi:10.37863/umzh.v75i5.7075

V. Priyadharshini Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, India
M. Nalliah Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, India

DOI: https://doi.org/10.37863/umzh.v75i5.7075

Keywords: Distance antimagic graphs, local distance antimagic chromatic number, star and double star graphs.

Abstract

UDC 519.17

Let $G=(V,E)$ be a graph on $p$ vertices with no isolated vertices. A bijection $f$ from $V$ to $ \{1,2,3,\ldots ,p\}$ is called a local distance antimagic labeling if, for any two adjacent vertices $u$ and $v,$ we receive distinct weights (colors), where a vertex $x$ has the weight $w(x)=\displaystyle\sum\nolimits_{v\epsilon N(x)} f(v).$ The local distance antimagic chromatic number $\chi_{lda}(G)$ is defined as the least number of colors used in any local distance antimagic labeling of $G.$ We determine the local distance antimagic chromatic number for the disjoint union of $t$ copies of stars and double stars.

References

S. Arumugam, D. Froncek, N. Kamatchi, Distance magic graphs — a survey, J. Indones. Math. Soc., Spec. Ed., 11–26 (2011). DOI: https://doi.org/10.22342/jims.0.0.15.11-26

S. Arumugam, N. Kamatchi, On $(a, d)$-distance antimagic graphs, Australas. J. Combin., 54, 279–287 (2012).

S. Arumugam, K. Premalatha, M. Bača, A. Semaničová-Fev{n}ovčiková, Local antimagic vertex coloring of a graph, Graphs and Combin., 33, 275–285 (2017). DOI: https://doi.org/10.1007/s00373-017-1758-7

M. Bača, M. Miller, Super edge-antimagic graphs: a wealth of problems and solutions, Brown Walker Press, Boca Raton (2008).

J. Bensmail, M. Senhaji, K. Szabo Lyngsie, On a combination of the 1-2-3 conjecture and the antimagic labelling conjecture, Discrete Math. Theor. Comput. Sci., 19, № 1 (2017).

G. Chartrand, L. Lesniak, Graphs and digraphs, 4th ed., Chapman and Hall, CRC (2005).

C. Colbourn, J. Dinitz, The CRC handbook of combinatorial designs, Chapman & Hall/CRC, Boca Raton (2007). DOI: https://doi.org/10.1201/9781420010541

N. J. Cutinhoa, S. Sudha, S. Arumugam, Distance antimagic labelings of Cartesian product of graphs, AKCE Int. J. Graphs and Comb., 17, № 3, 940–942 (2020).

D. Froncek, Handicap distance antimagic graphs and incomplete tournaments, AKCE Int. J. Graphs and Comb., 10, № 2, 119–127 (2013).

T. Divya, S. D. Yamini, Local distance antimagic vertex coloring of graphs; https://arxiv.org/abs/2106.01833v1.

J. A. Gallian, A dynamic survey of graph labeling, Electron. J. Combin. (2019). DOI: https://doi.org/10.37236/27

N. Hartsfield, G. Ringel, Pearls in graph theory, Acad. Press, INC, Boston (1994).

A. K. Handaa, A. Godinhoa, T. Singh, S. Arumugam, Distance antimagic labeling of join and corona of two graphs, AKCE Int. J. Graphs and Comb., 172–177 (2017). DOI: https://doi.org/10.1016/j.akcej.2017.04.003

A. K. Handa, A. Godinho, T. Singh, Distance antimagic labeling of the ladder graph, Electron. Notes Discrete Math., 63, 317–322 (2017). DOI: https://doi.org/10.1016/j.endm.2017.11.028

J. Haslegrave, Proof of a local antimagic conjecture, Discrete Math. Theor. Comput. Sci., 20, № 1 (2018).

N. Kamatchi, S. Arumugam, Distance antimagic graphs, J. Combin. Math. and Combin. Comput., 84, 61–67 (2013).

M. Miller, C. Rodger, R. Simanjuntak, Distance magic labelings of graphs, Australas. J. Combin., 28, 305–315 (2003).ň

M. Bača, A. Semaničová-Feňovčiková, Tao-Ming Wang, Local antimagic chromatic number for copies of graphs, Mathematics, 9, 1–12 (2021). DOI: https://doi.org/10.3390/math9111230

M. Nalliah, Antimagic labeling of graphs and digraphs, Ph. D. Thesis, Kalasalingam Univ., TamilNadu, India (2013).

V. Priyadharshini, M. Nalliah, Local distance antimagic chromatic number for the union of complete bipartite graphs, Tamkang J. Math. (to appear).

K. A. Sugeng, D. Froncek, M. Miller, J. Ryan, J. Walker, On distance magic labeling of graphs, J. Combin. Math. and Combin. Comput., 71, 39–48 (2009).

M. F. Semeniuta, $(a,d)$-Distance antimagic labeling of some types of graphs, Cybernet. and Systems Anal., 52, № 6, 950–959 (2016). DOI: https://doi.org/10.1007/s10559-016-9897-z

S. Shaebani, On local antimagic chromatic number of graphs, J. Algebraic Systems, 7, № 2, 245–256 (2020).

R. Shankar, M. Nalliah, Local vertex antimagic chromatic number of some wheel related graphs, Proyecciones, 41, № 1, 319–334 (2022). DOI: https://doi.org/10.22199/issn.0717-6279-4420

S. K. Patel, J. Vasava, Some results on $(a,d)$-distance antimagic labeling, Proyecciones, 39, № 2, 361–381 (2020). DOI: https://doi.org/10.22199/issn.0717-6279-2020-02-0022