On rainbow antimagic coloring and local edge antimagic coloring of graphs

Source abstract

<p>Let <em>G</em> be a connected graph of order <em>n</em>. Let <em>f</em>: <em>V</em>(<em>G</em>) → {1,2,...,<em>n</em>} be a bijection and for every <em>uv</em> ∈ <em>E</em>(<em>G</em>) consider <em>w</em><sub>f</sub>(<em>uv</em>) = <em>f</em>(<em>u</em>) + <em>f</em>(<em>v</em>) as a coloring of the edge. For a pair of vertices <em>u</em> and <em>v</em>, they are connected by a rainbow path if there exists a path from <em>u</em> to <em>v</em> such that the edges have pairwise distinct colors from <em>w</em><sub>f</sub>. The bijection <em>f</em>: <em>V</em>(<em>G</em>) → {1,2,...,<em>n</em>} is a <em>rainbow antimagic coloring</em> if for every two vertices there exists a rainbow path. Meanwhile, that bijection <em>f</em>: <em>V</em>(<em>G</em>) → {1,2,...,<em>n</em>} is <em>local edge antimagic coloring</em> if every two adjacent edges have distinct weights. The <em>rainbow antimagic connection number</em> <em>rac</em>(<em>G</em>) and <em>local edge antimagic chromatic number</em> <em>χ</em>'<sub>lea</sub>(<em>G</em>) is the minimum number of distinct edge weights over all rainbow antimagic coloring and local edge antimagic coloring, respectively.</p><p>We investigate the relationship between <em>rac</em>(<em>G</em>) and <em>χ</em>'<sub>lea</sub>(<em>G</em>). We prove <em>χ</em>'<sub>lea</sub>(<em>G</em>) ≤ <em>rac</em>(<em>G</em>) for all graphs and provide conditions for equality. Graphs with diameter at most 2 or satisfying <em>rac</em>(<em>G</em>) = Δ(G) achieve equality. We construct a family <em>A</em><sub>d</sub> with arbitrarily large diameter <em>d</em> where Δ(<em>A</em><sub>d</sub>) = <em>χ</em>'<sub>lea</sub>(<em>A</em><sub>d</sub>) = <em>rac</em>(<em>A</em><sub>d</sub>), and a family <em>H</em><sub>n</sub> = <em>K</em><sub>n,n</sub> - n<em>K</em><sub>2</sub> of diameter 3 where <em>χ</em>'<sub>lea</sub>(<em>H</em><sub>n</sub>) = rac(<em>H</em><sub>n</sub>) = 2<em>n</em>-3 > Δ(H<sub>n</sub>). These results present hints to the full characterization of graphs <em>G</em> with <em>χ</em>'<sub>lea</sub>(<em>G</em>) = <em>rac</em>(<em>G</em>).</p>