On Twin Intersection Chromatic Indices of Double Stars and Complete Graphs

Let G be a connected graph of order not less than three, let k≥2 be a positive integer, let [k] = {1, 2,..., k}, let P([k]) be the collection of all subsets of [k], and let f: E(G) → P([k])\\{0} be a proper edge coloring of G. For any vertex v of G, let f\' be a vertex coloring of G induced by ƒ defined as the intersection of the colors of all the edges of G incident with v. If f\' is proper, then ƒ is a twin intersection k-edge coloring of G. The smallest integer k for which G admits a twin intersection k-edge coloring is called the twin intersection chromatic index of G, denoted by X(G). In this paper, we fully determine the twin intersection chromatic index of double stars and provide some upper bounds by restricted join, thus determining the twin intersectin k-edge colorings of complete graphs.