The $k$-proper index of graphs

TL;DR

This paper introduces the concept of the $k$-proper index of graphs, exploring its properties, calculating it for specific graphs, and characterizing graphs with extremal $k$-proper index values.

Contribution

It defines the $k$-proper index, determines it for certain graphs, and constructs graphs with specified $k$-proper and $k$-rainbow indices, including extremal cases.

Findings

Determined $k$-proper indices for some special graphs.

Constructed graphs with prescribed $px_k(G)$ and $rx_k(G)$ values.

Characterized graphs with $k$-proper index $n-1$ and $n-2$.

Abstract

A tree $T$ in an edge-colored graph is a \emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2\leq k\leq n$. For a vertex set $S\subseteq V(G)$, a tree containing the vertices of $S$ in $G$ is called an \emph{$S$-tree}. An edge-coloring of $G$ is called a \emph{$k$-proper coloring} if for every set $S$ of $k$ vertices in $G$, there exists a proper $S$-tree in $G$. The \emph{$k$-proper index} of a nontrivial connected graph $G$, denoted by $px_k(G)$, is the smallest number of colors needed in a $k$-proper coloring of $G$. In this paper, some simple observations about $px_k(G)$ for a nontrivial connected graph $G$ are stated. Meanwhile, the $k$-proper indices of some special graphs are determined, and for every pair of positive integers $a$, $b$ with $2\leq a\leq b$, a connected graph $G$ with $px_k(G)=a$ and $rx_k(G)=b$ is constructed for each integer $k$ with $3\leq k\leq n$. Also, the graphs with $k$-proper index $n-1$ and $n-2$ are respectively characterized.