The $k$-proper index of complete bipartite and complete multipartite graphs

Wenjing Li, Xueliang Li, Jingshu Zhang

Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\to \{1,2,\dots ,t\}$,$t\in \mathbb{N}$, where adjacent edges may be colored with the same co…