On $(a,d)$-edge local antimagic coloring number of graphs

Authors: RAJKUMAR SUNDARAMOORTHY, NALLIAH MOVIRI CHETTIAR

Abstract: For any graph $G=(V,E),$ the order and size of G are $p$ and $q$. A bijection $l$ from $V(G)$ to $\{1,2,..,p\}$ is called $(a,d)$-edge local antimagic labeling if for any two adjacent edges are not received the same edge-weight (color) and the set of all edge-weights are formed an arithmetic progression $\{a,a+d,a+2d,\dots,a+(c-1)d\}$, for some integers $a,d>0$ and $c$ is the number of distinct colors used in the proper coloring.} An edge-weight (color) $w(uv)$ is the sum of two end vertices labels, $w(uv)=f(u)+f(v),uv\in E(G).$ The $(a,d)$-edge local antimagic coloring number is the least color (edge-weight) used in any $(a,d)$-edge local antimagic labeling. In the present study, we introduce a new type of labeling and a parameter, also we obtain the $(a,d)$-edge local antimagic coloring number for {paths and wheel graph $W_n$,$n=3,4,5$. Moreover, we obtain an upper bound of the $(a,d)$-edge local antimagic coloring number for wheel $W_n,n\ge 6$.

Keywords: $(a,d)$-edge local antimagic labeling, $(a,d)$-edge local antimagic coloring number, paths and wheel graph

Full Text: PDF