On Rainbow Connection

Abstract

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In this paper we prove several non-trivial upper bounds for $rc(G)$, as well as determine sufficient conditions that guarantee $rc(G)=2$. Among our results we prove that if $G$ is a connected graph with $n$ vertices and with minimum degree $3$ then $rc(G) < 5n/6$, and if the minimum degree is $\delta$ then $rc(G) \le {\ln \delta\over\delta}n(1+o_\delta(1))$. We also determine the threshold function for a random graph to have $rc(G)=2$ and make several conjectures concerning the computational complexity of rainbow connection.

Rainbow vertex connection of digraphs

“…Since ⌈ n−1 k ⌉ = ⌈ n k ⌉ if and only if n ≡ 1 (mod k), we instantly have → rvc(D) = ⌈ n k ⌉ − 1 by (4) and (5). To prove the second part, it suffices to show that → srvc(D) ≥ ⌈ n k ⌉ for n sufficiently large, by (4). Suppose that we have a vertex-colouring of D, using at most ⌈ n k ⌉ − 1 colours.…”

The rainbow connection of a graph is (at most) reciprocal to its minimum degree

“…If true, notice that for graphs with a linear minimum degree n, this implies that rc(G) is at most C / . However, the result from [5] does not even guarantee the weaker claim that rc(G) is a constant. This was proved recently by Chakraborty et al in [3].…”

“…Caro et al [5] observed that rc(G) can be bounded by a function of (G), the minimum degree of G. They have proved that if (G) ≥ 3 then rc(G) ≤ n where <1 is a constant and n = |V(G)|. They conjecture that = 3 / 4 suffices and prove that <5 / 6 (a solution to this conjecture was recently announced by Zsolt Tuza).…”

“…The diameter of this graph, as well as its rainbow connection, is n−3. More generally, it is proved in [5] that if = (G) then rc(G) ≤ (ln / )n(1+o (1)). An easier non-asymptotic bound rc(G) ≤ n(4 ln +3) / is also proved there.…”

“…We note that the constant 20 obtained by our proof is not optimal and can be slightly improved with additional effort. However, by the construction from [5] one cannot expect to replace C by a constant smaller than 3.…”