Conflict-free connection of trees
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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 χ_cf(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(P_n)=_2 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 2^k-1 vertices is k-1. At last, we construct some tree classes which are k-cfc-critical for every positive integer k.
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