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- W2803856394 abstract "A tree $T$ in an edge-colored graph is called a {it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2leq k leq n$. For $Ssubseteq V(G)$ and $|S| ge 2$, an $S$-tree is a tree containing the vertices of $S$ in $G$. Suppose ${T_1,T_2,ldots,T_ell}$ is a set of $S$-trees, they are called emph{internally disjoint} if $E(T_i)cap E(T_j)=emptyset$ and $V(T_i)cap V(T_j)=S$ for $1leq ineq jleq ell$. For a set $S$ of $k$ vertices of $G$, the maximum number of internally disjoint $S$-trees in $G$ is denoted by $kappa(S)$. The $kappa$-connectivity $kappa_k(G)$ of $G$ is defined by $kappa_k(G)=min{kappa(S)mid S$ is a $k$-subset of $V(G)}$. For a connected graph $G$ of order $n$ and for two integers $k$ and $ell$ with $2le kle n$ and $1leq ell leq kappa_k(G)$, the emph{$(k,ell)$-proper index $px_{k,ell}(G)$} of $G$ is the minimum number of colors that are needed in an edge-coloring of $G$ such that for every $k$-subset $S$ of $V(G)$, there exist $ell$ internally disjoint proper $S$-trees connecting them. In this paper, we show that for every pair of positive integers $k$ and $ell$ with $k ge 3$, there exists a positive integer $N_1=N_1(k,ell)$ such that $px_{k,ell}(K_n) = 2$ for every integer $n ge N_1$, and also there exists a positive integer $N_2=N_2(k,ell)$ such that $px_{k,ell}(K_{m,n}) = 2$ for every integer $n ge N_2$ and $m=O(n^r) (r ge 1)$. In addition, we show that for every $p ge csqrt[k]{frac{log_a n}{n}}$ ($c ge 5$), $px_{k,ell}(G_{n,p})le 2$ holds almost surely, where $G_{n,p}$ is the Erdos-Renyi random graph model." @default.
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- W2803856394 date "2016-06-13" @default.
- W2803856394 modified "2023-09-27" @default.
- W2803856394 title "The $(k,ell)$-proper index of graphs" @default.
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