Let $G$ be an edge-colored connected graph, a path (trail, walk) of $G$ is said to be a proper-path (trail, walk) if any two adjacent edges of it are colored distinctly. If there is a proper-path (trail, walk) between each pair of different vertices of $G,$ then $G$ is called proper-path (trail, walk) connected. The edge-coloring which makes $G$ proper-path (trail, walk) connected is called a proper-path (trail, walk) coloring. The minimum number of colors required in a proper-path (trail, walk) coloring is referred to as the proper-path (trail, walk) connection number of $G$. In J. Bang-Jensen, T. Bellitto and A. Yeo, Proper-walk connection number of graphs, J. Graph Theory 96(2020) 137–159, the authors investigated the graphs with proper-walk connection number $k$ and suggested to study the stretch of proper-walk coloring. In this note, we consider the stretch of proper-walk (path, trail) coloring and present some tight upper bounds.

让 $G$ 是一个边色的连通图，一条路径（小径，步行） $G$如果它的任何两个相邻边缘都有明显的颜色，则被称为一条正确的路径（小径，步行）。如果在的每对不同的顶点之间都有一条正确的路径（小径，步行）$G，$ 然后 $G$称为连接的正确路径（小径，步行）。边缘着色$G$连接的正确路径（步道，步道）称为适当路径（步道，步道）着色。适当路径（步道，步行）着色所需的最小颜色数称为适当路径（步道，步行）着色的连接数。$G$。在J. Bang-Jensen，T。Bellitto和A. Yeo的“图的正确行走连接数”，《 J。图论》 96（2020）137–159中，作者研究了具有正确行走连接数的图$ķ$并建议研究适当的步法着色的范围。在本说明中，我们考虑了适当的步行（路径，步道）着色的范围，并提出了一些严格的上限。