Large Rainbow Matchings in Edge-Colored Graphs Alexandr Kostochka, - - PowerPoint PPT Presentation

Introduction Set-up Case Analysis Conclusion

Large Rainbow Matchings in Edge-Colored Graphs

Alexandr Kostochka, Matthew Yancey1 May 13, 2011

1Department of Mathematics, University of Illinois, Urbana, IL 61801.

E-mail: yancey1@illinois.edu.

Introduction Set-up Case Analysis Conclusion

Rainbow Matchings in Graphs

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. r = rm(G) = number of edges in largest rainbow matching in G

A B C

Introduction Set-up Case Analysis Conclusion

Rainbow Matchings in Graphs

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. r = rm(G) = number of edges in largest rainbow matching in G

A B C

Introduction Set-up Case Analysis Conclusion

Rainbow Matchings in Graphs

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. r = rm(G) = number of edges in largest rainbow matching in G

A B C

r = 3

Introduction Set-up Case Analysis Conclusion

Rainbow Matchings in Graphs

A rainbow subgraph of an edge-colored graph is a subgraph whose edges have distinct colors. r = rm(G) = number of edges in largest rainbow matching in G

A B C

r = 3

Introduction Set-up Case Analysis Conclusion

Color Degree

For v ∈ V (G), ˆ d(v) is the number of distinct colors on the edges incident to v. k = ˆ δ(G) = min ˆ d(v)

A B C

Introduction Set-up Case Analysis Conclusion

Color Degree

For v ∈ V (G), ˆ d(v) is the number of distinct colors on the edges incident to v. k = ˆ δ(G) = min ˆ d(v)

A B C

Introduction Set-up Case Analysis Conclusion

Color Degree

For v ∈ V (G), ˆ d(v) is the number of distinct colors on the edges incident to v. k = ˆ δ(G) = min ˆ d(v)

A B C

d(A) = 3, ˆ d(A) = 2.

Introduction Set-up Case Analysis Conclusion

Color Degree

For v ∈ V (G), ˆ d(v) is the number of distinct colors on the edges incident to v. k = ˆ δ(G) = min ˆ d(v)

A B C

d(A) = 3, ˆ d(A) = 2. d(B) = 4, ˆ d(B) = 3.

Introduction Set-up Case Analysis Conclusion

Color Degree

For v ∈ V (G), ˆ d(v) is the number of distinct colors on the edges incident to v. k = ˆ δ(G) = min ˆ d(v)

A B C

d(A) = 3, ˆ d(A) = 2. d(B) = 4, ˆ d(B) = 3. d(C) = 4, ˆ d(C) = 2

Introduction Set-up Case Analysis Conclusion

Color Degree

For v ∈ V (G), ˆ d(v) is the number of distinct colors on the edges incident to v. k = ˆ δ(G) = min ˆ d(v)

A B C

d(A) = 3, ˆ d(A) = 2. d(B) = 4, ˆ d(B) = 3. d(C) = 4, ˆ d(C) = 2 k = 2

Introduction Set-up Case Analysis Conclusion

Color Degree

For v ∈ V (G), ˆ d(v) is the number of distinct colors on the edges incident to v. k = ˆ δ(G) = min ˆ d(v)

A B C

d(A) = 3, ˆ d(A) = 2. d(B) = 4, ˆ d(B) = 3. d(C) = 4, ˆ d(C) = 2 k = 2

Introduction Set-up Case Analysis Conclusion

Early Results

Theorem (Li and Wang, 2008) r ≥ 5k−3

12

• .

Theorem (Li and Wang, 2008) For k ≥ 3 and G bipartite, r ≥ 2k

3

• .

Conjecture (Li and Wang, 2008) For k ≥ 4, r ≥ k

2

• .

Introduction Set-up Case Analysis Conclusion

Tight Examples

k = 3 r = 1 k = 2 bipartite r = 1 Kn k = n - 1 r = n/2 or (n-1)/2

Introduction Set-up Case Analysis Conclusion

Further Results

Theorem (Li and Xu, 2007) If G is a properly colored complete graph other than K4, then r ≥ k

2

• .

Theorem (LeSaulnier, Stocker, Wegner, and West, 2010) r ≥ k

2

• and r ≥

k

2

• if any of the following are true

G is triange-free G is properly colored and n = k + 2 n ≥ 3(k−1)

2

Introduction Set-up Case Analysis Conclusion

Our Results

Theorem (Kostochka and Y, 2011+) If k ≥ 4, then r ≥ k

2

• Theorem (Kostochka and Y, 2011+)

If G is triangle-free, then r ≥ 2k

3

• .

Introduction Set-up Case Analysis Conclusion

Notation

1

u

... ...

2

u

i

u

r

u

1

v

2

v

i

v

r

v

1

w

2

w

i

w

p

w M H E Ei e e e e

1 2 i

r

r = k - 1 2 p = n - k + 1 edge has color i

ei

Introduction Set-up Case Analysis Conclusion

Notation

Let φ be an ordering on the vertices such that φ(u1) < φ(v1) < φ(u2) < φ(v2) < · · · < φ(ur) < φ(vr) An important edge, e = wz, is an edge with color not in M and with w ∈ H and z ∈ V (M) such that φ(z) is the minimum of all edges incident to w with the same color as wz.

Introduction Set-up Case Analysis Conclusion

The Inequality

number of important edges out of M = number of important edges out of H ≥ p(k − r) = (k + 1)p 2 On average the vertices in M are incident to more than p

2

important edges.

Introduction Set-up Case Analysis Conclusion

The Inequality

number of important edges out of M = number of important edges out of H ≥ p(k − r) = (k + 1)p 2 On average the vertices in M are incident to more than p

2

important edges. Lemma (LeSaulnier, Stocker, Wegner, and West, 2010) Each ei is incident to at most p + 1 important edges. Furthermore, if ei is incident to p + 1 important edges, then Ei has Configuration A or Configuration B.

Introduction Set-up Case Analysis Conclusion

The Inequality

number of important edges out of M = number of important edges out of H ≥ p(k − r) = (k + 1)p 2 On average the vertices in M are incident to more than p

2

important edges. Lemma (LeSaulnier, Stocker, Wegner, and West, 2010) Each ei is incident to at most p + 1 important edges. Furthermore, if ei is incident to p + 1 important edges, then Ei has Configuration A or Configuration B.

Introduction Set-up Case Analysis Conclusion

p + 1 Configurations

... ...

e

1

e2 ei er M H

... ...

e

1

e2 ei er M H

Configuration A Configuration B p = 3 u v

i

i

Introduction Set-up Case Analysis Conclusion

Special Vertices

A special vertex is a vertex v with d(v) = n − 1, ˆ d(v) = k, and

• ne color is incident to it n − k times (all other colors are incident

to it once). If Ei has Configuration A then vi is special.

Introduction Set-up Case Analysis Conclusion

Special Vertices

A special vertex is a vertex v with d(v) = n − 1, ˆ d(v) = k, and

• ne color is incident to it n − k times (all other colors are incident

to it once). If Ei has Configuration A then vi is special. Let v be a special vertex. The main color of v is the color that is repeated on the edges incident to v. A main edge of v is an edge incident to v colored the main color of v.

Introduction Set-up Case Analysis Conclusion

Special Vertices

A special vertex is a vertex v with d(v) = n − 1, ˆ d(v) = k, and

• ne color is incident to it n − k times (all other colors are incident

to it once). If Ei has Configuration A then vi is special. Let v be a special vertex. The main color of v is the color that is repeated on the edges incident to v. A main edge of v is an edge incident to v colored the main color of v.

Introduction Set-up Case Analysis Conclusion

Main Edges

We will assume that G is a minimal counter-example to the theorem, and that M contains the most main edges out of all maximum rainbow matchings in G. Lemma If Ei has Configuration A, then ui is special with main color i.

Introduction Set-up Case Analysis Conclusion

Main Edges

We will assume that G is a minimal counter-example to the theorem, and that M contains the most main edges out of all maximum rainbow matchings in G. Lemma If Ei has Configuration A, then ui is special with main color i. Proof. Suppose not. vi is special with a main color that is important, so it can not be i. Since ui is not special with main color i, edge ei is not a main edge. Replace ei with one of the main edges of vi. This is still a rainbow matching of same size, and now has more main edges, which contradicts our choice of M.

Introduction Set-up Case Analysis Conclusion

Main Edges

We will assume that G is a minimal counter-example to the theorem, and that M contains the most main edges out of all maximum rainbow matchings in G. Lemma If Ei has Configuration A, then ui is special with main color i. Proof. Suppose not. vi is special with a main color that is important, so it can not be i. Since ui is not special with main color i, edge ei is not a main edge. Replace ei with one of the main edges of vi. This is still a rainbow matching of same size, and now has more main edges, which contradicts our choice of M.

Introduction Set-up Case Analysis Conclusion

The Two Cases

Let a be the number of times Configuration A occurs and b be the number of times Configuration B occurs. Case 1: a > 0 We will assume that E1 has Configuration A. Consider edges u1ui for i such that Ei has Configuration A or B. We will attempt to replace e1 and ei with u1ui and important edges of v1 and vi to prove that M was not a maximum rainbow matching.

Introduction Set-up Case Analysis Conclusion

The Two Cases

Let a be the number of times Configuration A occurs and b be the number of times Configuration B occurs. Case 1: a > 0 We will assume that E1 has Configuration A. Consider edges u1ui for i such that Ei has Configuration A or B. We will attempt to replace e1 and ei with u1ui and important edges of v1 and vi to prove that M was not a maximum rainbow matching.

Introduction Set-up Case Analysis Conclusion

5-Alternating Paths

...

e

1

eh ei er M H

If edge u1ui has color 1, i, or a color not in M, then we generate a contradiction to either the maximality of M or minimality of G.

Introduction Set-up Case Analysis Conclusion

5-Alternating Path Plus a 2-Alternating Path

If the color of edge u1ui matches the color of edge eh in M.

e

1

eh ei er M H

Introduction Set-up Case Analysis Conclusion

5-Alternating Path Plus a 2-Alternating Path

If the color of edge u1ui matches the color of edge eh in M.

e

1

eh ei er M H

Try to replace e1, ei, and eh with u1ui and important edges adjacent to v1, vi, and vh.

Introduction Set-up Case Analysis Conclusion

5-Alternating Path Plus a 2-Alternating Path

If the color of edge u1ui matches the color of edge eh in M.

e

1

eh ei er M H

Try to replace e1, ei, and eh with u1ui and important edges adjacent to v1, vi, and vh.

Introduction Set-up Case Analysis Conclusion

Conclusion of Case 1

pk + 1 2 ≤ number of important edges out of H = number of important edges out of M ≤ (a + b)(p + 1) + (a + b − 1)2 + (k − 1 2 − 2a − 2b + 1)p And Case 1 is done!

Introduction Set-up Case Analysis Conclusion

Case 2

We no longer have special vertices to use. However, we know that p = 3. (k − 1) + p = n n = k + 2

Introduction Set-up Case Analysis Conclusion

Case 2

We no longer have special vertices to use. However, we know that p = 3. (k − 1) + p = n n = k + 2 Every vertex is adjacent to n − 2 distinct colors. Every vertex is ”like” a special vertex!

Introduction Set-up Case Analysis Conclusion

Case 2

We no longer have special vertices to use. However, we know that p = 3. (k − 1) + p = n n = k + 2 Every vertex is adjacent to n − 2 distinct colors. Every vertex is ”like” a special vertex!

Introduction Set-up Case Analysis Conclusion

Open Problem

Conjecture (Li and Wang, 2008) For bipartite graphs, r ≥ k − 1 if k is even and r ≥ k if k is odd. If true, this would be a generalization of H. J. Ryser’s conjecture (1967) for the maximum size of a transveral in a latin square.

Introduction Set-up Case Analysis Conclusion

• M. Kano and X. Li, Monochromatic and heterochromatic subgraphs

in edge-colored graphs–a survey. Graphs Combin. 24 (2008), 237-263.

• T. D. LeSaulnier, C. Stocker, P. S. Wegner, and D. B. West,

Rainbow Matching in Edge-Colored Graphs. Electron. J. Combin. 17 (2010), Paper #N26.

• H. Li and G. Wang, Color degree and heterochromatic matchings in

edge-colored bipartite graphs. Util. Math. 77 (2008), 145–154.

• X. Li and Z. Xu, On the existence of a rainbow 1-factor in proper

coloring of K (r)

rn . arXiv:0711.2847 [math.CO] 19 Nov 2007.

• G. Wang and H. Li, Heterochromatic matchings in edge-colored
• graphs. Electron. J. Combin. 15 (2008), Paper #R138.