An edge-colored graph G is conflict-free connected if any two of its vertices are connected by a path, which contains a color used on exactly one of * the minimum degree δ(G) ≥ max{3, n−4 5 }, then cf c(G) = 2. The bound on the minimum degree is best possible. Next, we prove that, if G is a connected non-complete graph of order n ≥ 33 with C(G) being a linear forest and with d(x) + d(y) ≥ 2n−9 5 for each pair of two nonadjacent vertices x, y of V (G), then cf c(G) = 2. Both bounds, on the order n and the degree sum, are tight. Moreover, we prove several results concerning relations between degree conditions on G and the number of cut edges in G.
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 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.