Conflict-free connection of trees

An edge-colored graph G is conflict-free connected if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is defined as the smallest number of colors that are required in order to make G conflict-free connected. A coloring of vertices of a hypergraph H = (V, E) is called conflict-free if each hyperedge e of H has a vertex of unique color that does not get repeated in e. The smallest number of colors required for such a coloring is called the conflict-free chromatic number of H, and is denoted by χcf (H). In this paper, we study the conflict-free connection coloring of trees, which is also the conflict-free coloring of edge-path hypergraphs of trees. We first prove that for a tree T of order n, cfc(T ) ≥ cfc(Pn) = ⌈log2 n⌉, and this completely confirms the conjecture of Li and Wu. We then present a sharp upper bound for the conflict-free connection number of trees by a simple algorithm. Furthermore, we show that the conflict-free connection number of the binomial tree with 2k−1 vertices is k − 1. At last, we construct some tree classes which are k-cfc-critical for every positive integer k.