Rainbow connection and forbidden subgraphs

An Updated Survey on Rainbow Connections of Graphs- A Dynamic Survey

“…For |F | = 1, it was shown in [66] that the answer for the above problem is the same as in Theorem 2.11, that is, the only such graph H = P 3 . For |F | = 2, the following result shows that this situation is the same as in Theorem 2.12.…”

“…In [66], the authors gave a complete answer for |F | ∈ {1, 2} by showing the following two theorems. The first theorem characterizes all connected graphs H such every connected…”

“…Although, in general, there is no upper bound on rc(G) in terms of diam(G), and, in bridgeless graphs, rc(G) can be quadratic in terms of diam(G) as shown before, it turns out that forbidden subgraph conditions can remarkably lower the upper bound on rc(G). In [66], Holub et al considered the following problem. Problem 2.10.…”

“…[66] Let H be a connected graph. Then there is a constant k H such that every connected H-free graph G satisfies rc(G) ≤ diam(G) + k H , if and only if H = P 3 .…”

3-Rainbow Index and Forbidden Subgraphs

“…In [7], Holub et al considered the question: For which families F of connected graphs, a connected F -free graph G satisfies rc(G) ≤ diam(G) + C F , where C F is a constant (depending on F ), and they gave a complete answer for |F | ∈ {1, 2} in the following two results (where N denotes the net, a graph obtained by attaching a pendant edge to each vertex of a triangle). Let k ≥ 3 be a positive integer.…”