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, cf c(T ) ≥ cf c(P n ) = ⌈log 2 n⌉, 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 cf c-critical, and prove that if a tree T is cf c-critical, then the conflict-free connection coloring of T is equivalent to the edge ranking of T .

(Strong) conflict-free connectivity: Algorithm and complexity

“…Hence, they got a tight upper bound for the conflict-free vertex-connection number of connected graphs of order n. In the same paper, Li and Wu posed, as a conjecture, and Chang et al in [9] verified the following result.…”

“…One can easily see that the conflict-free chromatic number of the hypergraph H is just the conflict-free connection number of T . For more results we refer to [7,8,9,12]. Nevertheless, most of them are about the graph structural characterizations.…”

Hardness results for three kinds of colored connections of graphs

“…This path is called a conflict-free path, and this coloring is called a conflict-free connection coloring of G. The conflict-free connection number of a connected graph G, denoted by cf c(G), is defined as the smallest number of colors needed to color the edges of G such that G is conflict-free connected. More results can be found in [6,7,8].…”

“…In this paper, we prove that to compute each of pc(G), cf c(G) and mc(G) of a graph is NP-hard. This solves a long standing problem in this field, asked in many talks of workshops and papers; see [20,21,25,7,8,18].…”

Conflict-free (vertex)-connection numbers of graphs with small diameters

“…The conflict-free connection number of a connected graph G, denoted by cf c(G), is defined as the smallest number of colors required to make G conflict-free connected. There are many results on this topic, for more details, please refer to [2,3,4,5,6]. It is easy to see that 1 ≤ cf c(G) ≤ n − 1 for a connected graph G.…”