Distance proper connection of graphs

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper connected if every pair of distinct vertices of $G$ are connected by $k$ pairwise internally vertex-disjoint distance $\ell$-proper paths in $G$. For a $k$-connected graph $G$, the minimum number of colors needed to make $G$ $(k,\ell)$-proper connected is called the $(k,\ell)$-proper connection number of $G$ and denoted by $pc_{k,\ell}(G)$. In this paper, we prove that $pc_{1,2}(G)\leq 5$ for any $2$-connected graph $G$. Considering graph operations, we find that $3$ is a sharp upper bound for the $(1,2)$-proper connection number of the join and the Cartesian product of almost all graphs. In addition, we find some basic properties of the $(k,\ell)$-proper connection number and determine the values of $pc_{1,\ell}(G)$ where $G$ is a traceable graph, a tree, a complete bipartite graph, a complete multipartite graph, a wheel, a cube or a permutation graph of a nontrivial traceable graph.

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