Conflict-free connection of trees

TL;DR

This paper investigates the conflict-free connection coloring of trees, establishing bounds, exact values for special trees, and linking critical trees to edge rankings, thus advancing understanding of conflict-free colorings in graph theory.

Contribution

It proves a lower bound for the conflict-free connection number of trees, provides a sharp upper bound via a simple algorithm, and links critical trees to edge rankings.

Findings

Lower bound for cfc(T) as  cfc(P_n)

Exact cfc value for binomial trees

Characterization of cfc-critical 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 $n$, $cfc(T)\geq cfc(P_n)=\lceil \log_{2} n\rceil$, which 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 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 $T$.