Introduction Set-up Case Analysis Conclusion Large Rainbow Matchings in Edge-Colored Graphs Alexandr Kostochka, Matthew Yancey 1 May 13, 2011 1 Department of Mathematics, University of Illinois, Urbana, IL 61801. E-mail: yancey1@illinois.edu.
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
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
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 )
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) � 5 k − 3 � r ≥ . 12 Theorem (Li and Wang, 2008) � 2 k � For k ≥ 3 and G bipartite, r ≥ . 3 Conjecture (Li and Wang, 2008) � k � For k ≥ 4 , r ≥ . 2
Introduction Set-up Case Analysis Conclusion Tight Examples Kn k = 2 k = 3 bipartite k = n - 1 r = 1 r = n/2 or (n-1)/2 r = 1
Introduction Set-up Case Analysis Conclusion Further Results Theorem (Li and Xu, 2007) If G is a properly colored complete graph other than K 4 , then � k � r ≥ . 2 Theorem (LeSaulnier, Stocker, Wegner, and West, 2010) � k � k � � r ≥ and r ≥ if any of the following are true 2 2 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+) � k � If k ≥ 4 , then r ≥ 2 Theorem (Kostochka and Y, 2011+) � 2 k � If G is triangle-free, then r ≥ . 3
Introduction Set-up Case Analysis Conclusion Notation E H M w 1 r = k - 1 e v u 1 1 1 w 2 2 v u e 2 2 2 p = n - k + 1 w ... i u v e i i i E i e i edge has color i ... w u v e p r r r
Introduction Set-up Case Analysis Conclusion Notation Let φ be an ordering on the vertices such that φ ( u 1 ) < φ ( v 1 ) < φ ( u 2 ) < φ ( v 2 ) < · · · < φ ( u r ) < φ ( v r ) 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 e i is incident to at most p + 1 important edges. Furthermore, if e i is incident to p + 1 important edges, then E i 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 e i is incident to at most p + 1 important edges. Furthermore, if e i is incident to p + 1 important edges, then E i has Configuration A or Configuration B.
Introduction Set-up Case Analysis Conclusion p + 1 Configurations M H M H e e 1 1 e 2 e 2 ... ... v e i u e i i i ... ... e r e r Configuration B Configuration A p = 3
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 one color is incident to it n − k times (all other colors are incident to it once). If E i has Configuration A then v i 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 one color is incident to it n − k times (all other colors are incident to it once). If E i has Configuration A then v i 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 one color is incident to it n − k times (all other colors are incident to it once). If E i has Configuration A then v i 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 E i has Configuration A, then u i 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 E i has Configuration A, then u i is special with main color i. Proof. Suppose not. v i is special with a main color that is important, so it can not be i . Since u i is not special with main color i , edge e i is not a main edge. Replace e i with one of the main edges of v i . 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 E i has Configuration A, then u i is special with main color i. Proof. Suppose not. v i is special with a main color that is important, so it can not be i . Since u i is not special with main color i , edge e i is not a main edge. Replace e i with one of the main edges of v i . 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 E 1 has Configuration A. Consider edges u 1 u i for i such that E i has Configuration A or B. We will attempt to replace e 1 and e i with u 1 u i and important edges of v 1 and v i to prove that M was not a maximum rainbow matching.
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