This paper studies the problem of proper-walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph i.e. a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with $k$ colours for every possible value of $k$.

中文翻译：

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图的正确行走连接数

本文研究了正确游走连接数的问题：给定一个无向连通图，我们的目标是用尽可能少的颜色给它的边着色，以便在图的每一对顶点之间存在一个正确着色的游走，即游走不连续使用相同颜色的两条边。这个问题已经在几类图上解决了，但在一般情况下仍然是开放的。我们确定问题总是可以在多项式时间内以图的大小解决，并且我们提供了可以与 $k$ 的每个可能值的 $k$ 颜色正确连接的图的表征。