Large Rainbow Matchings in Edge-Colored Graphs

Transcription

1 Large Rainbow Matchings in Edge-Colored Graphs Alexandr Kostochka, Matthew Yancey 1 May 13, Department of Mathematics, University of Illinois, Urbana, IL yancey1@illinois.edu.

2 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

3 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

4 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

5 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

6 Color Degree For v V (G), ˆd(v) is the number of distinct colors on the edges incident to v. k = ˆδ(G) = min ˆd(v)

7 Color Degree For v V (G), ˆd(v) is the number of distinct colors on the edges incident to v. k = ˆδ(G) = min ˆd(v) A B C

8 Color Degree For v V (G), ˆd(v) is the number of distinct colors on the edges incident to v. k = ˆδ(G) = min ˆd(v) A B C d(a) = 3, ˆd(A) = 2.

9 Color Degree For v V (G), ˆd(v) is the number of distinct colors on the edges incident to v. k = ˆδ(G) = min ˆd(v) A B C d(a) = 3, ˆd(A) = 2. d(b) = 4, ˆd(B) = 3.

10 Color Degree For v V (G), ˆd(v) is the number of distinct colors on the edges incident to v. k = ˆδ(G) = min ˆd(v) A B C d(a) = 3, ˆd(A) = 2. d(b) = 4, ˆd(B) = 3. d(c) = 4, ˆd(C) = 2

11 Color Degree For v V (G), ˆd(v) is the number of distinct colors on the edges incident to v. k = ˆδ(G) = min ˆd(v) A B C d(a) = 3, ˆd(A) = 2. d(b) = 4, ˆd(B) = 3. d(c) = 4, ˆd(C) = 2 k = 2

12 Color Degree For v V (G), ˆd(v) is the number of distinct colors on the edges incident to v. k = ˆδ(G) = min ˆd(v) A B C d(a) = 3, ˆd(A) = 2. d(b) = 4, ˆd(B) = 3. d(c) = 4, ˆd(C) = 2 k = 2

13 Early Results Theorem (Li and Wang, 2008) r 5k 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..

14 Tight Examples k = 3 r = 1 k = 2 bipartite r = 1 K n k = n - 1 r = n/2 or (n-1)/2

15 Further Results Theorem (Li and Xu, 2007) If G is a properly colored complete graph other than K 4, 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

16 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.

17 Notation M E H E i u 1 u 2 u i u r e 1 v 1 v 2 2 e e e i r v i v r w i w 2 w 1 w p edge r = k p = n - k + 1 e i has color i

18 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.

19 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.

20 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.

21 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.

22 p + 1 Configurations M H M H e 1 e 1 e 2 e 2 u i e i v i e i e r e r Configuration A Configuration B p = 3

23 Special Vertices A special vertex is a vertex v with d(v) = n 1, ˆd(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.

24 Special Vertices A special vertex is a vertex v with d(v) = n 1, ˆd(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.

25 Special Vertices A special vertex is a vertex v with d(v) = n 1, ˆd(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.

26 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.

27 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.

28 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.

29 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.

30 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.

31 5-Alternating Paths M H e 1 e h e i... e r If edge u 1 u i 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.

32 5-Alternating Path Plus a 2-Alternating Path If the color of edge u 1 u i matches the color of edge e h in M. M H e 1 e h e i e r

33 5-Alternating Path Plus a 2-Alternating Path If the color of edge u 1 u i matches the color of edge e h in M. M H e 1 e h e i e r Try to replace e 1, e i, and e h with u 1 u i and important edges adjacent to v 1, v i, and v h.

34 5-Alternating Path Plus a 2-Alternating Path If the color of edge u 1 u i matches the color of edge e h in M. M H e 1 e h e i e r Try to replace e 1, e i, and e h with u 1 u i and important edges adjacent to v 1, v i, and v h.

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

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

37 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!

38 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!

39 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.

40 M. Kano and X. Li, Monochromatic and heterochromatic subgraphs in edge-colored graphs a survey. Graphs Combin. 24 (2008), 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), X. Li and Z. Xu, On the existence of a rainbow 1-factor in proper coloring of K (r) rn. arxiv: [math.co] 19 Nov G. Wang and H. Li, Heterochromatic matchings in edge-colored graphs. Electron. J. Combin. 15 (2008), Paper #R138.