Applied Mathematics and Computation ( IF 3.5 ) Pub Date : 2021-04-13 , DOI: 10.1016/j.amc.2021.126240 Zhongmei Qin , Junxue Zhang
Let be an edge-colored connected graph, a path (trail, walk) of 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 then is called proper-path (trail, walk) connected. The edge-coloring which makes 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 . 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 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.
中文翻译:
图的正确游动着色的极端拉伸
让 是一个边色的连通图,一条路径(小径,步行) 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) 然后 称为连接的正确路径(小径,步行)。边缘着色连接的正确路径(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。。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。