Some results on strong conflict-free connection number of graphs

Abstract

A path in an edge-colored graph is called a conflict-free path if there exists a color used on only one of its edges. An edge-colored graph is called conflict-free connected if there is a conflict-free path between each pair of distinct vertices. We call the graph [Formula: see text] strongly conflict-free connectedif there exists a conflict-free path of length [Formula: see text] for every two vertices [Formula: see text]. The strong conflict-free connection number of a connected graph [Formula: see text], denoted by [Formula: see text], is defined as the smallest number of colors that are required to make [Formula: see text] strongly conflict-free connected. The authors in [M. Ji, X. Li and X. Zhu, (Strong) Conflict-free connectivity: Algorithm and complexity, Theor. Comput. Sci. 804 (2020) 72–80] showed that the problem of deciding whether [Formula: see text] [Formula: see text] for a given graph [Formula: see text] is NP-complete. In this paper, we give mainly some bounds of strong conflict-free connection number of graphs and illustrate the difference between [Formula: see text] and [Formula: see text],[Formula: see text].