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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 $2leq kleq n$. For a vertex set $Ssubseteq V(G)$, 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 $2leq aleq b$, a connected graph $G$ with $px_k(G)=a$ and $rx_k(G)=b$ is constructed for each integer $k$ with $3leq kleq n$. Also, the graphs with $k$-proper index $n-1$ and $n-2$ are respectively characterized.
Let $chi_k(G)$ denote the minimum number of colors needed to color the edges of a graph $G$ in a way that the subgraph spanned by the edges of each color has all degrees congruent to $1 pmod k$. Scott [{em Discrete Math. 175}, 1-3 (1997), 289--291] p
Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)tomathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)tomathbb{Z}$ defined by $f^+(uv)=f(u)+f(v)$, and $f^-:E
The Wiener index of a graph $G$, denoted $W(G)$, is the sum of the distances between all pairs of vertices in $G$. E. Czabarka, et al. conjectured that for an $n$-vertex, $ngeq 4$, simple quadrangulation graph $G$, begin{equation*}W(G)leq begin{cas
A connected graph $G$ is said to be $k$-connected if it has more than $k$ vertices and remains connected whenever fewer than $k$ vertices are deleted. In this paper, for a connected graph $G$ with sufficiently large order, we present a tight sufficie
Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $ngeqslant 10$ ar