Theory and Applications of Graphs Theory and Applications of Graphs

In this note, we provide a sharp upper bound on the rainbow connection number of tournaments of diameter 2. For a tournament T of diameter 2, we show 2 ≤ −→ rc ( T ) ≤ 3. Furthermore, we provide a general upper bound on the rainbow k -connection number of tournaments as a simple example of the probabilistic method. Finally, we show that an edge-colored tournament of k th diameter 2 has rainbow k -connection number at most approximately k 2 .


Introduction
The concept of rainbow connection was first introduced by Chartrand et al. in [3].A path in an edge-colored graph is called rainbow if no two edges in the path receive the same color.The rainbow connection number of a graph is the minimum number of colors needed to color the edges of the graph so that there is a rainbow path between every pair of vertices.This and the more general rainbow k-connection number have been heavily studied in recent years in [2,3,4,5,7,8,10] and many other works.In particular, see [9] for a survey of results in the area.
A tournament T is an oriented complete graph.We consider only k-strongly connected (or simply k-strong) tournaments, meaning that there are k internally disjoint directed paths from each vertex to every other vertex.A directed path between two vertices in an edgecolored tournament is called rainbow if no two edges have the same color within the path.If there is a directed rainbow path between every pair of vertices in a graph, then the coloring is called rainbow connected.The smallest number of colors needed for a tournament to be rainbow connected is called the (directed) rainbow connection number, denoted − → rc(T ).The diameter d of a tournament is the largest, over all ordered pairs of vertices, number of edges in the shortest path between the two vertices.
In [6], the following theorem was proven.
Theorem 1 (Dorbec et al. [6]).For any tournament T of diameter d, The authors noted that d + 2 may not be the best upper bound.We believe that a (d + 1)-coloring is possible, at least in some cases.Indeed, we show that for tournaments of diameter 2, this improved upper bound holds.The proof of this result is provided in Section 2.
More generally, we initiate the study of the rainbow k-connection number of a tournament.An edge-colored tournament is called rainbow k-connected if, between every pair of vertices, there is a set of k internally disjoint rainbow paths.The rainbow k-connection number of a tournament, denoted − → rc k (T ), is then the minimum number of colors needed to produce a rainbow k-connected coloring of the tournament T .To state our next result, we let the k-total-diameter, denoted d k (T ), be the maximum (over all pairs of vertices) of the smallest number of edges in a set of k internally disjoint paths between the vertices.
Theorem 3. Given an integer k ≥ 2 and a tournament T of order n with . The proof of Theorem 3 is an easy application of the probabilistic method and is presented in Section 3.
Next we define some more notation.Say a set of k internally disjoint paths from a vertex x to a vertex y is minimum if the longest path in the set is as short as possible, over all such sets of paths.Let the k th diameter denote the maximum length, over all pairs of vertices u, v, of the longest path in a minimum set of Note that the 1 st diameter is simply the diameter of the graph.Also note that the k th diameter is at least d k (T ) k .Our final result considers tournaments with small k th diameter and provides a bound on the rainbow connection number. 2 Proof of Theorem 2 The sharpness of the upper bound is given by the following example.Let A be a directed triangle and let u and v be single vertices.Direct all edges from v to A, from A to u and from u to v. Any 2-coloring of this graph must color two edges of A with a single color.This induces a directed monochromatic P 3 .Let a 1 be the initial vertex of this P 3 and let a 3 be the terminal vertex and note that the edge between a 1 and a 3 is directed from a 3 to a 1 .See Figure 1.The only possible rainbow path from a 1 to a 3 must pass through u and v, meaning that it must use 3 different colors.Thus, this graph has diameter 2 but rainbow connection number 3. Larger graphs with the same property can be built by replacing vertices with directed triangles and blowing up edges in the natural way.
We now prove Theorem 2.

A u v
Figure 1: A tournament T with diameter 2 and rainbow connection number 3.
Proof.Let T be a tournament of diameter 2. Let a → b → c be a shortest path from a vertex a to another vertex c.Let A 1 denote the out-neighborhood of a and let Note that A i is the set of vertices at distance i from a and, in particular, b ∈ A 1 and c ∈ A 2 .
Color all edges from a to A 1 with color 1 (red in Figure 2).All edges from vertices in A 1 to vertices in A 2 have color 2 (blue), and all edges from vertices in A 2 to the vertex a have color 3 (green).All edges from vertices in A 2 to vertices in A 1 also have color 3. Finally, all edges within the same set, either A 1 or A 2 , have color 1.This coloring is similar to the one used by Dorbec et al. in [6] to prove Theorem 1.In order to show that this coloring is rainbow connected, we consider cases based on the location of two selected vertices x and y and find rainbow paths between them.
If x = a, we trivially find a rainbow path to y for any choice of y ∈ A 1 since A 1 is the out-neighborhood of a.If y ∈ A 2 , then by construction, there is a rainbow path containing some vertex w ∈ A 1 such that x → w → y with colors 1 and 2 respectively.
If y = a and x ∈ A 2 , the result is again trivial since a is an out-neighbor of every vertex in A 2 .Also, if x ∈ A 1 , then again there is a rainbow path of length 2, namely x → w → y for some w ∈ A 2 using colors 2 and 3.
If x ∈ A 2 and y ∈ A 1 , then, by construction, there is a rainbow path x → a → y using colors 3 and 1, respectively.
Finally, suppose x ∈ A 1 and y ∈ A 2 .If the edge x → y is in E(T ), then there is a trivially rainbow path of length 1.Since the diameter is 2, there exists a vertex with x → w → y.Regardless of the location of w, this path is rainbow by construction.More specifically, if w ∈ A 1 , then the path x → w → y uses colors 1 and 2, respectively.If w ∈ A 2 , then the path x → w → y exists uses colors 2 and 1, respectively.Hence, every tournament with diameter 2 has rainbow connection number at most 3.

Proof of Theorem 3
For this proof, we use the probabilistic method as described by Alon and Spencer in [1].This bound is likely far from the best possible, particularly when n is much bigger than d, and we make little effort to optimize it.
We now prove Theorem 3.
Proof.Consider a tournament on n vertices and set c = . Label the vertices of T with {v 1 , v 2 , . . ., v n }.Randomly color the edges of T using c colors.Let X i,j be an indicator variable which takes the value 1 if there is no set of k internally disjoint rainbow paths from v i to v j .Since there is a set of such paths on at most d edges, we compute the expectation of X i,j to be By linearity of expectation, if we set X = i,j X i,j , we get we see that E(X) < 1 so, by the probabilistic method, there is a coloring of T with c colors that is rainbow k-connected.

Proof of Theorem 4
For a tournament T of k th diameter 2, we use the following coloring.Select a k-subset of vertices A := {v 1 , v 2 , . . ., v k }.Now, let the set of all out-neighbors of A be called A 1 and color all edges of the form A → A 1 with the color C 1 and edges of the form A 1 → A with color C 2 .Let N ′ i ⊆ A 1 be the out-neighborhood of v i for each 1 ≤ i ≤ k.Define sets for all 1 ≤ i ≤ k.We use one distinct color C N i on all edges within each set N i for all i for a total of k colors.Use at most 2 k 2 distinct colors to color the remaining edges of A 1 such that edges of the form N i → N j have a different color from those of the form N j → N i for all i = j.This uses a total of 2 k 2 + k colors to color A 1 .Let A 2 be the set of all remaining vertices so that A 2 is in the out-neighborhood of A 1 and A is in the out-neighborhood of A 2 .It should be noted that A 2 = ∅ is allowed.Color all edges of the form A 1 → A 2 with color C 2 and the edges from A 2 → A with color C 3 .All edges from A to A 2 have color C 1 and all edges from A 2 to A 1 have color C 3 .The

Question 1 .
For each diameter d, is d + 1 or d + 2 the sharp upper bound on − → rc(T ) where T has diameter d.

Theorem 4 .
A strongly connected tournament T of k th diameter 2 has − → rc k (T ) ≤ 3+k+2 k 2 .The proof of Theorem 4 is presented in Section 4. This naturally leads to the following problem.Problem 1. Produce sharp bounds on − → rc k (T ) in terms of the k th diameter of T .

Figure 2 :
Figure 2: Coloring of the tournament.