On SD-Prime Labelings of Some One Point Unions of Gears

Let G=(V(G),E(G)) be a simple, finite and undirected graph of order n. Given a bijection f:V(G)→{1,…,n}, and every edge uv in E(G), let S=f(u)+f(v) and D=|f(u)−f(v)|. The labeling f induces an edge labeling f′:E(G)→{0,1} such that for an edge uv in E(G), f′(uv)=1 if gcd(S,D)=1, and f′(uv)=0 otherwise. Such a labeling is called an SD-prime labeling if f′(uv)=1 for all uv∈E(G). We provide SD-prime labelings for some one point unions of gear graphs.