On local antimagic vertex coloring of corona products related to friendship and fan graph

Abstract

Let <em>G</em>=(<em>V</em>,<em>E</em>) be connected graph. A bijection <em>f </em>: <em>E</em> → {1,2,3,..., |<em>E</em>|} is a local antimagic of <em>G</em> if any adjacent vertices <em>u,v</em> ∈ <em>V</em> satisfies <em>w</em>(<em>u</em>)≠ <em>w</em>(<em>v</em>), where <em>w</em>(<em>u</em>)=∑<sub>e∈E(u) </sub><em>f</em>(<em>e</em>), <em>E</em>(<em>u</em>) is the set of edges incident to <em>u</em>. When vertex <em>u</em> is assigned the color <em>w</em>(<em>u</em>), we called it a local antimagic vertex coloring of <em>G</em>. A local antimagic chromatic number of <em>G</em>, denoted by <em>χ</em><sub>la</sub>(<em>G</em>), is the minimum number of colors taken over all colorings induced by the local antimagic labeling of <em>G</em>. In this paper, we determine the local antimagic chromatic number of corona product of friendship and fan with null graph on <em>m</em> vertices, namely, <em>χ</em><sub>la</sub>(<em>F</em><sub>n</sub> ⊙ \overline{K_m}) and <em>χ</em><sub>la</sub>(<em>f</em><sub>(1,n)</sub> ⊙ \overline{K_m}).

On local antimagic chromatic number of a corona product graph

“…In [8], the authors studied χ la (f n • O m ) and χ la (F n • O m ) for n ≥ 2 and m ≥ 1. We note that there are inconsistencies in the notations of f n and F n used.…”

A note on local edge antimagic chromatic number of graphs

“…They also found some bounds of local antimagic chromatic number for trees. There are other studies about local antimagic chromatic number which involves complete full t-ary trees [4], wheels and helms [7], corona products related to friendship and fan graph [11], graphs amalgamation [12], generalized friendship graphs [14], and lexicographic product graphs [13]. In addition, Haslegrave [10] has proven that every connected graphs other than K 2 has a local antimagic labeling.…”