Mathematica Bohemica
Gary Chartrand; Garry L. Johns; Kathleen A. McKeon; Ping Zhang
Rainbow connection in graphs
Mathematica Bohemica, Vol. 133 (2008), No. 1, 85–98
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133 (2008) MATHEMATICA BOH EMICA No. 1, 85–98
RAINBOW CONNECTION IN GRAPHS
Gary Chartrand, Kalamazoo, Garry L. Johns, Saginaw Valley,
Kathleen A. McKeon, New London, Ping Zhang, Kalamazoo
(Received July 31, 2006)
Abstract. Let G be a nontrivial connected graph on which is defined a coloring c: E(G) →
{1, 2, . . . , k}, k ∈
N
, of the edges of G, where adjacent edges may be colored the same. A
path P in G is a rainbow path if no two edges of P are colored the same. The graph G
is rainbow-connected if G contains a rainbow u − v path for every two vertices u and v of
G. The minimum k for which there exists such a k- edge coloring is the rainbow connection
number rc(G) of G. If for every pair u, v of distinct vertices, G contains a rainbow u − v
geodesic, then G is strongly rainbow-connected. The minimum k for which there exists
a k-edge coloring of G that results in a strongly rainbow-connected graph is called the
strong rainbow connection number src(G) of G. Thus rc(G) 6 src(G) for every nontrivial
connected graph G. Both rc(G) and src(G) are determined for all complete multipartite
graphs G as well as other classes of graphs. For every pair a, b of integers with a > 3 and
b > (5a − 6)/3, it is shown that there exists a connected graph G such that rc(G) = a and
src(G) = b.
Keywords: edge coloring, rainbow coloring, strong rainbow coloring
MSC 2000 : 05C15, 05C38, 05C40
1. Introduction
Let G be a nontrivial connected graph on which is defined a coloring c : E(G) →
{1, 2, . . . , k}, k ∈
N
, of the edges of G, where adjacent edges may be colo red the
same. A u − v path P in G is a rainbow path if no two edges of P are c olored
the same. The graph G is rainbow-connected (with respect to c) if G c ontains a
rainbow u −v path for every two vertices u and v of G. In this case, the color ing c is
called a rainbow coloring of G. If k color s are used, then c is a rainbow k-coloring.
The minimum k for which there exists a rainbow k-coloring of the edges of G is the
rainbow connection number rc(G) of G. A rainbow colo ring of G using rc(G) colors
is called a minimum rainbow coloring of G.
85
Let c be a rainbow coloring of a connected graph G. For two vertices u and v of G,
a rainbow u −v geodesic in G is a rainbow u −v path of length d(u, v), where d(u, v)
is the distance between u and v (the length of a shortest u −v path in G). The graph
G is strongly rainbow-connected if G contains a rainbow u −v geodesic for e very two
vertices u and v of G. In this case, the color ing c is called a strong rainbow coloring
of G. The minimum k for which there exists a coloring c: E(G) → { 1, 2, . . . , k}
of the edges of G such that G is strongly rainbow-connected is the strong rainbow
connection number src(G) of G. A strong rainbow coloring of G using src(G) colors
is called a minimum strong rainbow coloring of G. Thus rc(G) 6 src(G) fo r every
connected gr aph G.
Since every coloring that assigns distinct colors to the edges of a connected graph
is b oth a rainbow coloring and a strong rainbow coloring, every connected graph is
rainbow-connected and strongly rainbow-connected with respect to some coloring of
the edges of G. Thus the rainbow connection numbers rc(G) and src(G) are defined
for every connected graph G. Furthermore, if G is a nontrivial co nnected graph of
size m whose diameter (the large st distance between two vertices of G) is diam(G),
then
(1) diam(G) 6 rc(G) 6 src(G) 6 m.
To illustrate these concepts, consider the Petersen graph P of Fig ure 1, where a
rainbow 3-coloring of P is also shown. Thus rc(P ) 6 3. On the other hand, if u and
v are two nona djac e nt vertices of P , then d(u, v) = 2 and so the length of a u − v
path is at leas t 2. Thus any rainbow coloring of P uses at leas t two colors and so
rc(P ) > 2. If P has a rainbow 2-coloring c, then there exist two adjacent edges of G
that are colored the same by c, say e = uv and f = vw are colored the same. Since
there is exa ctly one u − w path of length 2 in P , there is no ra inbow u − w path in
P , which is a contradiction. Therefore, rc(P ) = 3.
2
1
3
2
1
3
2
3
1
32
1 1
3 2
2
1
3
2
1
3
2
3
1
42
1 4
3 4
Figure 1. A rainbow 3-coloring and a strong rainbow 4-coloring of th e Petersen graph
86
Since rc(P ) = 3, it follows that src(P) > 3. Furthermore, since the edge chromatic
number of the Petersen graph is known to be 4, any 3-coloring c of the edges of P
results in two adjace nt edges uv and vw being assigned the same color. Since u, v, w
is the only u − w geodesic in P , the coloring c is not a s trong rainbow coloring.
Because the 4-c oloring of the e dges of P shown in Figure 2 is a strong rainbow
coloring, src(P) = 4.
As another example, consider the graph G of Figure 2(a), where a rainbow 4-
coloring c of G is also shown. In fact, c is a minimum rainbow coloring of G and so
rc(G) = 4, as we now verify.
u
u
1
v
1
v
v
3
u
3
u
2
v
2
1
4
2 1
4
3
2
4
3
(a)
u
u
1
v
1
v
v
3
u
3
u
2
v
2
1
2
3
3
2
(b)
Figure 2. A graph G with rc(G) = src(G) = 4
Since diam(G) > 3, it follows that rc(G) > 3. Assume, to the contrary, rc(G) = 3.
Then there exists a rainbow 3- c oloring c
′
of G. Since every u − v path in G has
length 3, at least one of the three u − v paths in G is a rainbow u − v path, say
u, u
1
, v
1
, v is a rainbow u − v path. We may assume that c
′
(uu
1
) = 1, c
′
(u
1
v
1
) = 2,
and c
′
(v
1
v) = 3. (See Figure 2(b).)
If x and y are two vertices in G such that d(x, y) = 2, then G contains exactly
one x −y path o f length 2, while all other x − y paths have length 4 or more. This
implies that no two adjacent edges can be colored the same. Thus we may assume,
without loss of generality, that c
′
(uu
2
) = 2 and c
′
(uu
3
) = 3. (See Figure 2(b).) Thus
{c
′
(vv
2
), c
′
(vv
3
)} = {1, 2}. If c
′
(vv
2
) = 1 a nd c
′
(vv
3
) = 2, then c
′
(u
2
v
2
) = 3 a nd
c
′
(u
3
v
3
) = 1. In this case, there is no rainbow u
1
− v
3
path in G. On the other
hand, if c
′
(vv
2
) = 2 and c
′
(vv
3
) = 1, then c
′
(u
2
v
2
) ∈ {1, 3} and c
′
(u
3
v
3
) = 2. If
c
′
(u
2
v
2
) = 1, then there is no rainbow u
2
−v
3
path in G; while if c
′
(u
2
v
2
) = 3, there
is no rainbow u
2
− v
1
path in G, a contradiction. Therefore, as claimed, rc(G) = 4.
Since 4 = rc(G) 6 src(G) for the graph G of Figure 2 and the rainbow 4-coloring
of G in Figure 2(a) is also a strong rainbow 4-coloring, src(G) = 4 as well.
If G is a nontrivial connected graph of size m, then we saw in (1) that diam(G) 6
rc(G) 6 src(G) 6 m. In the following result, it is determined which connected graphs
G attain the extreme values 1, 2 or m.
87
Proposition 1.1. Let G be a nontrivial connected graph of size m. Then
(a) src(G) = 1 if and only if G is a complete graph,
(b) rc(G) = 2 if and only if src(G) = 2,
(c) rc(G) = m if and only if G is a tree.
P r o o f. We firs t verify (a). If G is a complete graph, then the coloring that
assigns 1 to every edge of G is a strong rainbow 1-coloring of G and so src(G) = 1.
On the other hand, if G is not complete, then G contains two nonadjacent vertices
u and v. Thus each u −v geodesic in G has length at least 2 and so src(G) > 2.
To verify (b), first assume that rc(G) = 2 and so src(G) > 2 by (1). Since
rc(G) = 2, it follows that G has a rainbow 2-coloring, which implies that every two
nonadjacent vertices are connected by a rainbow path of length 2. Because such a
path is a geodesic, src(G) = 2. On the other hand, if src(G) = 2, then rc(G) 6 2 by
(1) aga in. Further more, since src(G) = 2, it follows by (a) that G is not complete
and so rc(G) > 2. Thus rc(G) = 2.
We now verify (c). Suppos e first that G is not a tree. Then G contains a cycle
C : v
1
, v
2
, . . . , v
k
, v
1
, where k > 3. Then the (m − 1)-co loring of the edges of G
that assig ns 1 to the edges v
1
v
2
and v
2
v
3
and ass igns the m −2 distinct colors from
{2, 3, . . . , m − 1} to the remaining m − 2 edges of G is a rainbow coloring. Thus
rc(G) 6 m − 1. Next, let G be a tree of size m. Assume, to the contrary, that
rc(G) 6 m − 1. Let c be a minimum rainbow coloring of G. Then there exist edges
e and f such that c(e) = c(f ). Assume, without loss generality, that e = uv and
f = xy and G contains a u − y path u, v, . . . , x, y. Then there is no rainbow u − y
path in G, which is a contradiction.
Proposition 1.1 also implies that the only connected graphs G for which rc(G) = 1
are the complete graphs and that the only connected graphs G of size m for which
src(G) = m are trees.
2. Some rainbow connection numbers of graphs
In this section, we determine the rainbow connection numbers of some well-known
graphs. We refer to the book [1] for graph-theoretical notation and terminolo gy not
described in this paper. We begin with cycles of order n. Since diam(C
n
) = ⌊n/2⌋,
it follows by (1) that src(C
n
) > rc(C
n
) > ⌊n/2⌋. This lower bound for rc(C
n
) and
src(C
n
) is near ly the exact value of these numbers.
88