Vertex‐Distinguishing and Sum‐Distinguishing Edge Coloring of Regular Graphs

ABSTRACT Given an integer , an edge‐‐coloring of a graph is an assignment of colors to the edges of such that no two adjacent edges receive the same color. A vertex‐distinguishing (resp. sum‐distinguishing) edge‐‐coloring of is an edge‐‐coloring such that for any two distinct vertices and , the set (resp. sum) of colors taken from all the edges incident with is different from that taken from all the edges incident with . The vertex‐distinguishing chromatic index (resp. sum‐distinguishing chromatic index), denoted (resp. ), is the smallest value such that has a vertex‐distinguishing edge‐‐coloring (resp. sum‐distinguishing edge‐‐coloring). Let be a ‐regular graph on vertices, where is even and sufficiently large. We show that if is arbitrarily close to from above, and if . Our first result strengthens a result of Balister et al. for such class of regular graphs, and our second result constitutes a significant advancement in the field of sum‐distinguishing edge coloring. To achieve these results, we introduce novel edge coloring results which may be of independent interest.