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

Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\rightarrow\{1,2,\dots,t\}$,$t\in\mathbb{N}$, where adjacent edges may be colored with the same color. A tree $T$ in $G$ is a \emph{proper tree} if no two adjacent edges of it are assigned the same color. Let $k$ be a fixed integer with $2\leq k\leq n$. For a vertex subset $S\subseteq V(G)$ with $|S|\geq 2$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$ . A \emph{$k$-proper coloring} of $G$ is an edge-coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a proper $S$-tree $T$ in $G$. The minimum number of colors that are needed in a $k$-proper coloring of $G$ is defined as the \emph{$k$-proper index} of $G$, denoted by $px_k(G)$. In this paper, we determine the 3-proper index of all complete bipartite and complete multipartite graphs and partially determine the $k$-proper index of them for $k\geq 4$.

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