Rainbow coloring explained

In graph theory, a path in an edge-colored graph is said to be rainbow if no color repeats on it. A graph is said to be rainbow-connected (or rainbow colored) if there is a rainbow path between each pair of its vertices. If there is a rainbow shortest path between each pair of vertices, the graph is said to be strongly rainbow-connected (or strongly rainbow colored).

Definitions and bounds

The rainbow connection number of a graph

G is the minimum number of colors needed to rainbow-connect G , and is denoted by rc (G) . Similarly, the strong rainbow connection number of a graph G is the minimum number of colors needed to strongly rainbow-connect G , and is denoted by src (G) . Clearly, each strong rainbow coloring is also a rainbow coloring, while the converse is not true in general.

It is easy to observe that to rainbow-connect any connected graph

G , we need at least diam (G) colors, where diam (G) is the diameter of G (i.e. the length of the longest shortest path). On the other hand, we can never use more than m colors, where m denotes the number of edges in G . Finally, because each strongly rainbow-connected graph is rainbow-connected, we have that diam (G) \leq rc (G) \leq src (G) \leq m .

The following are the extremal cases:

rc (G) = src (G) = 1 if and only if G is a complete graph.rc (G) = src (G) = m if and only if G is a tree.

The above shows that in terms of the number of vertices, the upper bound

rc (G) \leq n - 1 is the best possible in general. In fact, a rainbow coloring using n - 1 colors can be constructed by coloring the edges of a spanning tree of G in distinct colors. The remaining uncolored edges are colored arbitrarily, without introducing new colors. When G is 2-connected, we have that rc (G) \leq \lceil n/ 2 \rceil . Moreover, this is tight as witnessed by e.g. odd cycles.

Exact rainbow or strong rainbow connection numbers

The rainbow or the strong rainbow connection number has been determined for some structured graph classes:

rc (C _{n)} = src (C _{n)} = \lceil n/ 2 \rceil , for each integer n \geq 4 , where C _{n} is the cycle graph.rc (W _{n)} = 3 , for each integer n \geq 7 , and src (W _{n)} = \lceil n/ 3 \rceil , for n \geq 3 , where W _{n} is the wheel graph.

Complexity

The problem of deciding whether

rc (G) = 2 for a given graph G is NP-complete. Because rc (G) = 2 if and only if src (G) = 2 , it follows that deciding if src (G) = 2 is NP-complete for a given graph G .

Variants and generalizations

Chartrand, Okamoto and Zhang generalized the rainbow connection number as follows. Let

G be an edge-colored nontrivial connected graph of order n . A tree T is a rainbow tree if no two edges of T are assigned the same color. Let k be a fixed integer with 2 \leq k \leq n . An edge coloring of G is called a k -rainbow coloring if for every set S of k vertices of G , there is a rainbow tree in G containing the vertices of S . The k -rainbow index rx _{k(G)} of G is the minimum number of colors needed in a k -rainbow coloring of G . A k -rainbow coloring using rx _{k(G)} colors is called a minimum k -rainbow coloring. Thus rx _{2(G)} is the rainbow connection number of G .

Rainbow connection has also been studied in vertex-colored graphs. This concept was introduced by Krivelevich and Yuster.Here, the rainbow vertex-connection number of a graph

G , denoted by rvc (G) , is the minimum number of colors needed to color G such that for each pair of vertices, there is a path connecting them whose internal vertices are assigned distinct colors.

See also

• Rainbow matching

References

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