Conflict-free connection of trees

Abstract

We study the conflict-free connection coloring of trees, which is also the conflict-free coloring of the so-called edge-path hypergraphs of trees. We first prove that for a tree T of order n, $$ cfc (T)\ge cfc (P_n)=\lceil \log _{2} n\rceil $$ , which completely confirms the conjecture of Li and Wu [24]. 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 study trees which are $$ cfc $$ -critical, and prove that if a tree T is $$ cfc $$ -critical, then the conflict-free connection coloring of T is equivalent to the edge ranking of T.