Abstract
Let \(G=(V,E)\) be a connected graph with \(\left| V \right| =n\) and \(\left| E \right| = m.\) A bijection \(f:E \rightarrow \{1,2, \dots , m\}\) is called a local antimagic labeling if for any two adjacent vertices u and v, \(w(u)\ne w(v),\) where \(w(u)=\sum \nolimits _{e\in E(u)}{f(e)},\) and E(u) is the set of edges incident to u. Thus any local antimagic labeling induces a proper vertex coloring of G where the vertex v is assigned the color w(v). The local antimagic chromatic number \(\chi _{la}(G)\) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper we present several basic results on this new parameter.
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Acknowledgements
The authors are thankful to the reviewer for helpful suggestions which led to substantial improvement in the presentation of the paper. The second author is thankful to Kalasalingam University for providing University Research Fellowship.
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Arumugam, S., Premalatha, K., Bača, M. et al. Local Antimagic Vertex Coloring of a Graph. Graphs and Combinatorics 33, 275–285 (2017). https://doi.org/10.1007/s00373-017-1758-7
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DOI: https://doi.org/10.1007/s00373-017-1758-7