The $$(k,\ell )$$ ( k , ℓ ) -proper index of graphs

Abstract

A tree T in an edge-colored graph is called a 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 $$2\le k \le n$$ . For $$S\subseteq V(G)$$ and $$|S| \ge 2$$ , an S-tree is a tree containing the vertices of S in G. A set $$\{T_1,T_2,\ldots ,T_\ell \}$$ of S-trees is called internally disjoint if $$E(T_i)\cap E(T_j)=\emptyset $$ and $$V(T_i)\cap V(T_j)=S$$ for $$1\le i\ne j\le \ell $$ . 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 k-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 $$2\le k\le n$$ and $$1\le \ell \le \kappa _k(G)$$ , the $$(k,\ell )$$ -proper index $$px_{k,\ell }(G)$$ of G is the minimum number of colors that are required 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 $$k \ge 3$$ and $$\ell \le \kappa _k(K_{n,n})$$ , 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 there exists also 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 $$p \ge c\root k \of {\frac{\log _a n}{n}}$$ ( $$c \ge 5$$ ), $$px_{k,\ell }(G_{n,p})\le 2$$ holds almost surely, where $$G_{n,p}$$ is the Erdős–Renyi random graph model.