Graphs with Conflict-Free Connection Number Two

Abstract

An edge-colored graph G is conflict-free connected if every two of its vertices are connected by a path, which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph G, denoted by cfc(G), is the smallest number of colors needed in order to make G conflict-free connected. For a graph G,  let C(G) be the subgraph of G induced by its set of cut-edges. In this paper, we first show that, if G is a connected non-complete graph G of order $n\ge 9$ with C(G) being a linear forest and with the minimum degree $\delta (G)\ge \max \{3, \frac{n-4}{5}\}$, then $cfc(G)=2$. The bound on the minimum degree is best possible. Next, we prove that, if G is a connected non-complete graph of order $n\ge 33$ with C(G) being a linear forest and with $d(x)+d(y)\ge \frac{2n-9}{5}$ for each pair of two nonadjacent vertices x, y of V(G), then $cfc(G)=2$. 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.