Edge-disjoint rainbow spanning trees in
complete graphs.
James Carraher
University of Nebraska – Lincoln
[email protected]
Joint work with Stephen Hartke and Paul Horn
April 2013
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Rainbow Spanning Trees
Def. Let G be an edge-colored graph (not necessarily proper).
A rainbow spanning tree is a spanning tree T of G, where
each edge of T has a different color.
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Rainbow Spanning Trees
Def. Let G be an edge-colored graph (not necessarily proper).
A rainbow spanning tree is a spanning tree T of G, where
each edge of T has a different color.
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Previous Work
Brualdi and Hollingsworth asked how many edge-disjoint
rainbow spanning trees does an edge-colored graph Kn have?
What is an upper bound on the number of edge-disjoint
spanning trees in Kn ?
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Previous Work
Brualdi and Hollingsworth asked how many edge-disjoint
rainbow spanning trees does an edge-colored graph Kn have?
What is an upper bound on the number of edge-disjoint
spanning trees in Kn ?
n
Kn has 2 edges, each spanning tree has n − 1 edges, so
š 
there are at most 2n edge-disjoint spanning trees.
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Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
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Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
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Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
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Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
4 / 20
Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
This coloring of Kn has 3 edge-disjoint rainbow spanning trees.
4 / 20
Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
This coloring of Kn has 3 edge-disjoint rainbow spanning trees.
4 / 20
Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
This coloring of Kn has 3 edge-disjoint rainbow spanning trees.
4 / 20
Previous Work
Conjecture [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6,
and Kn is colored where each color class forms a perfect
matching, then Kn contains 2n edge-disjoint rainbow spanning
trees.
This coloring of Kn has 3 edge-disjoint rainbow spanning trees.
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Previous Work
Thm. [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6, and Kn
is colored where each color class is a perfect matching, then
Kn has at least 2 edge-disjoint rainbow spanning trees.
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Previous Work
Thm. [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6, and Kn
is colored where each color class is a perfect matching, then
Kn has at least 2 edge-disjoint rainbow spanning trees.
Thm. [Kaneko, Kano, Suzuki 2003] If Kn is properly
edge-colored with n ≥ 6, then Kn has at least 3 edge-disjoint
rainbow spanning trees.
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Previous Work
Thm. [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6, and Kn
is colored where each color class is a perfect matching, then
Kn has at least 2 edge-disjoint rainbow spanning trees.
Thm. [Kaneko, Kano, Suzuki 2003] If Kn is properly
edge-colored with n ≥ 6, then Kn has at least 3 edge-disjoint
rainbow spanning trees.
Conj. [Kaneko, Kano, Suzuki 2003] If šKnis properly
edge-colored with n ≥ 6, then Kn has 2n edge-disjoint
rainbow spanning trees.
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Previous Work
Thm. [Brualdi, Hollingsworth 1996] If n is even, n ≥ 6, and Kn
is colored where each color class is a perfect matching, then
Kn has at least 2 edge-disjoint rainbow spanning trees.
Thm. [Kaneko, Kano, Suzuki 2003] If Kn is properly
edge-colored with n ≥ 6, then Kn has at least 3 edge-disjoint
rainbow spanning trees.
Conj. [Kaneko, Kano, Suzuki 2003] If šKnis properly
edge-colored with n ≥ 6, then Kn has 2n edge-disjoint
rainbow spanning trees.
Thm. [Akabari, Alipour 2007] If Kn is edge-colored (not
necessarily proper) where each color is on at most 2n edges,
then Kn has at least 2 edge-disjoint rainbow spanning trees.
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Theorem
Thm. [C., Hartke, Horn 2013+] If Kn is edge-colored such that
each color is on at most 2n edges and n ≥ 600, 000, then G has
j
k
at least t = 874nlog n edge-disjoint rainbow spanning trees.
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Theorem
Thm. [C., Hartke, Horn 2013+] If Kn is edge-colored such that
each color is on at most 2n edges and n ≥ 600, 000, then G has
j
k
at least t = 874nlog n edge-disjoint rainbow spanning trees.
Our approach
Randomly construct t edge-disjoint subgraphs G1 , . . . , Gt .
Show that each subgraphs G contains a rainbow spanning
tree with high probability.
Show that the event that G1 , . . . , Gt simultaneously have a
rainbow spanning tree has positive probability.
Thus, G has at least t edge-disjoint rainbow spanning trees.
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Theorem
Thm. [C., Hartke, Horn 2013+] If Kn is edge-colored such that
each color is on at most 2n edges and n ≥ 600, 000, then G has
j
k
at least t = 874nlog n edge-disjoint rainbow spanning trees.
Our approach
Randomly construct t edge-disjoint subgraphs G1 , . . . , Gt .
Show that each subgraphs G contains a rainbow spanning
tree with high probability.
Show that the event that G1 , . . . , Gt simultaneously have a
rainbow spanning tree has positive probability.
Thus, G has at least t edge-disjoint rainbow spanning trees.
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Theorem
Construction of the subgraphs G1 , . . . , Gt . For each edge
uniformly at random pick a number in {1, . . . , t}. The edges
with label  are in the subgraph G .
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Theorem
Construction of the subgraphs G1 , . . . , Gt . For each edge
uniformly at random pick a number in {1, . . . , t}. The edges
with label  are in the subgraph G .
1
3
3
2
2
1
1
2
3
1
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Theorem
Construction of the subgraphs G1 , . . . , Gt . For each edge
uniformly at random pick a number in {1, . . . , t}. The edges
with label  are in the subgraph G .
1
1
1
1
7 / 20
Theorem
Construction of the subgraphs G1 , . . . , Gt . For each edge
uniformly at random pick a number in {1, . . . , t}. The edges
with label  are in the subgraph G .
2
2
2
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Theorem
Construction of the subgraphs G1 , . . . , Gt . For each edge
uniformly at random pick a number in {1, . . . , t}. The edges
with label  are in the subgraph G .
3
3
3
7 / 20
Theorem
Thm.[C. Hartke, Horn 2013+] If Kn is edge colored such that
each color class has at most 2n edges and n ≥ 600, 000, then
k
j
G has at least t = 874nlog n edge-disjoint rainbow spanning
trees.
Our approach
Randomly construct t edge-disjoint subgraphs G1 , . . . , Gt .
Show that each subgraphs G contains a rainbow spanning
tree with high probability.
Show that the event that each of G1 , . . . , Gt have a
rainbow spanning tree has positive probability.
Thus, G has at least t edge-disjoint rainbow spanning trees.
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Theorem
Thm.[Broersma and Li 1997; Schrijver 2003; Suzuki 2006]
A graph G has a rainbow spanning tree if and only if for every
partition π of the vertices of G into s parts, there are at least
s − 1 colors between the parts.
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Theorem
Thm.[Broersma and Li 1997; Schrijver 2003; Suzuki 2006]
A graph G has a rainbow spanning tree if and only if for every
partition π of the vertices of G into s parts, there are at least
s − 1 colors between the parts.
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Theorem
Thm.[Broersma and Li 1997; Schrijver 2003; Suzuki 2006]
A graph G has a rainbow spanning tree if and only if for every
partition π of the vertices of G into s parts, there are at least
s − 1 colors between the parts.
Proof ⇐ One proof uses the Matroid Intersection Theorem.
Take the graphic matroid whose independent sets are forests
of G, and the partition matroid whose independent sets take
at most one edge from each color class.
The largest independent set that is in both matroids is the
largest rainbow forest contained in G.
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Proof of Theorem
Show that each subgraphs G contains a rainbow spanning
tree with high probability.
The proof is broken up into cases based on how many parts s
the partition π of V(G) has.
Case 1 When s = n or s = n − 1.
Case 2 When .53n ≤ s ≤ n − 2.
Case 3 When 2 ≤ s ≤ .53n.
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Proof of Theorem
Show that each subgraphs G contains a rainbow spanning
tree with high probability.
The proof is broken up into cases based on how many parts s
the partition π of V(G) has.
Case 1 When s = n or s = n − 1.
Suppose each color shows up at least 4n times. We want to
show that each subgraph G has at least one color from every
color class.
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Proof of Theorem
Bernstein’s Inequality
Sse X are iid Bernoulli random
P
variables, X = X . Then
λ2
‚
P [X ≥ E[X] + λ] ≤ exp −
2(E[X] + λ/ 3)
λ2
‚
P [X ≤ E[X] − λ] ≤ exp −
.
Œ
Œ
2E[X]
Union Sum Bound

P
r
[
=1

A  ≤
r
X
P[A ]
=1
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Proof of Theorem
Lemma For a fixed G and color class Cj
–
P |E(G ) ∩ Cj | ≤
|Cj |
t
r
−
n
3
t
™
log n ≤
1
n3
.
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Proof of Theorem
Lemma For a fixed G and color class Cj
–
P |E(G ) ∩ Cj | ≤
|Cj |
t
r
−
n
3
t
™
log n ≤
1
n3
.
Proof. By Bernstein’s Inequality:


r
–
™
|Cj |
n
−3 nt log n

− 3 log n ≤ exp 
P |E(G ) ∩ Cj | ≤
|C |
t
t
2 tj
‚
Œ
−3n log n
1
= exp
=
2 2n
n3
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Proof of Theorem
Lemma For a fixed G and color class Cj
–
P |E(G ) ∩ Cj | ≤
|Cj |
t
r
−
n
3
t
™
log n ≤
1
n3
.
Proof. By Bernstein’s Inequality:


r
–
™
|Cj |
n
−3 nt log n

− 3 log n ≤ exp 
P |E(G ) ∩ Cj | ≤
|C |
t
t
2 tj
‚
Œ
−3n log n
1
= exp
=
2 2n
n3
By the union sum bound with probability 1 − 1n each color
class has at least one edge in every subgraph G1 , . . . , Gt .
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Proof of Theorem
Lemma For a fixed G and color class Cj
–
P |E(G ) ∩ Cj | ≤
|Cj |
t
r
−
n
3
t
™
log n ≤
1
n3
.
Proof of Case 1
If s = n, then there are at least n − 1 different colors between
the parts by the previous Lemma.
When s = n − 1 there is at most one edge inside the parts, so
there are at least n − 2 different colors between the parts by
the Lemma.
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Proof of Theorem
Case 2. Suppose .53n ≤ s ≤ n − 2.
Fix G and a partition π into s parts.
G
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Proof of Theorem
Case 2. Suppose .53n ≤ s ≤ n − 2.
Fix G and a partition π into s parts.
Kn
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Proof of Theorem
Case 2. Suppose .53n ≤ s ≤ n − 2.
Fix G and a partition π into s parts.
Kn with no
blue or green
edges
The probability the edges not colored blue or green are not in
€
Šk
G is 1 − 1t .
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Proof of Theorem
Case 2. Suppose .53n ≤ s ≤ n − 2.
Fix G and a partition π into s parts.
Kn with no
blue or green
edges
The probability the edges not colored blue or green are not in
€
Šk
G is 1 − 1t .
We show the probability remains small when bounding over
all different colors between the parts, all partitions of size at
least .53n, and all t subgraphs G .
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Proof of Theorem
Case 3 Suppose 2 ≤ s ≤ .53n.
With high
Æ probability each color is represented at most
n
+ 4 nt log n times. (Bernstein)
2t
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Proof of Theorem
Case 3 Suppose 2 ≤ s ≤ .53n.
With high
Æ probability each color is represented at most
n
+ 4 nt log n times. (Bernstein)
2t
If there are fewer than s − 1 colors between the parts, then an
upper bound on the number of edges between the parts is
given by
r
Œ
‚
n
n
(s − 2)
+4
log n .
2t
t
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Proof of Theorem
Lemma For S ⊆ V(G), the number of edges between S and S
in G is at least
È
|S|(n − |S|)
6|S|(n − |S|)
−
min{|S|, n − |S|} log n.
t
t
S
S
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Proof of Theorem
Lemma For S ⊆ V(G), the number of edges between S and S
in G is at least
È
6|S|(n − |S|)
|S|(n − |S|)
−
min{|S|, n − |S|} log n.
t
t
A lower bound on the number of edges between the parts of
π is
X
1
ƒ (|P |)
2 π={P1 ,...,Ps }
q
(n−)
6(n−)
where ƒ () =
−
min{, n − } log n.
t
t
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Proof of Theorem
Using convexity arguments, the sum is minimized when the
first part has size n − s + 1 and the rest have size 1. I.e.
1
X
2 π={P1 ,...,Ps }
where ƒ () =
(n−)
t
ƒ (|P |) ≥
−
q
1
2
((s − 1)ƒ (1) + ƒ (n − s + 1))
6(n−)
t
min{, n − } log n.
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Proof of Theorem
An upper bound
Æ on thenumber of edges between the parts
n
is (s − 2) 2t + 4 nt log n .
A lower bound on the number of edges between the parts is
1
((s − 1)ƒ (1) + ƒ (n − s + 1)).
2
Using algebra we can show
1
2
‚
((s − 1)ƒ (1) + ƒ (n − s + 1)) > (s − 2)
n
2t
r
+4
Œ
n
t
log n
This implies there must be more than s − 2 colors between the
parts. Thus, with high probability G has a rainbow spanning
tree.
„
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Conclusion
Thm.[C. Hartke, Horn 2013+] If Kn is edge-colored such that
each color class has at most 2n edges and n ≥ 65, 000, then G
j
k
has at least t = 874nlog n edge-disjoint rainbow spanning
trees.
Thm.[Horn 2013+] There exists constants ε, n0 > 0 such that
for n > n0 and Kn is properly edge-colored such that each
color class forms a perfect matching, then Kn has at least εn
edge-disjoint rainbow spanning trees.
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Questions
Open Questions.
Conj.šIf Kn is properly edge-colored with n ≥ 6, then Kn has at
least 2n edge-disjoint rainbow spanning trees.
Can you improve the bound on the number of
edge-disjoint rainbow spanning trees?
How many edge-disjoint rainbow spanning trees can other
graphs contain?
20 / 20
Questions
Open Questions.
Conj.šIf Kn is properly edge-colored with n ≥ 6, then Kn has at
least 2n edge-disjoint rainbow spanning trees.
Can you improve the bound on the number of
edge-disjoint rainbow spanning trees?
How many edge-disjoint rainbow spanning trees can other
graphs contain?
Thank you
20 / 20