Long monochromatic paths and cycles in 2-edge-colored multipartite graphs Xujun Liu University of Illinois at Urbana-Champaign Joint work with J´ ozsef Balogh, Alexandr Kostochka and Mikhail Lavrov 31st Cumberland Conference on Combinatorics, Graph Theory and Computing May 18, 2019 Xujun Liu (UIUC) Long cycle May 18, 2019 1 / 23
Overview Introduction 1 Main results 2 Xujun Liu (UIUC) Long cycle May 18, 2019 2 / 23
Outline Introduction 1 Main results 2 Xujun Liu (UIUC) Long cycle May 18, 2019 3 / 23
Introduction For graphs G 0 , . . . , G k we write G 0 �→ ( G 1 , . . . , G k ) if for every k -coloring of the edges of G 0 , for some i ∈ [ k ] there will be a copy of G i with all edges of color i . The Ramsey number R k ( G ) is the minimum N such that K N �→ ( G 1 , . . . , G k ), where G 1 = . . . = G k = G . Xujun Liu (UIUC) Long cycle May 18, 2019 4 / 23
Introduction For graphs G 0 , . . . , G k we write G 0 �→ ( G 1 , . . . , G k ) if for every k -coloring of the edges of G 0 , for some i ∈ [ k ] there will be a copy of G i with all edges of color i . The Ramsey number R k ( G ) is the minimum N such that K N �→ ( G 1 , . . . , G k ), where G 1 = . . . = G k = G . Example R 2 ( K 3 ) = 6. Figure: R 2 ( K 3 ) = 6. Xujun Liu (UIUC) Long cycle May 18, 2019 4 / 23
Previous results In this talk, we denote P n the path on n vertices and M n the matching with n edges. A matching M in G is connected if all edges of M are in the same component of G . Xujun Liu (UIUC) Long cycle May 18, 2019 5 / 23
Previous results In this talk, we denote P n the path on n vertices and M n the matching with n edges. A matching M in G is connected if all edges of M are in the same component of G . � 3 n − 2 � Gerencs´ er and Gy´ arf´ as (1967): R 2 ( P n ) = . They actually 2 showed for positive integers k ≥ ℓ , R ( P k , P ℓ ) = k − 1 + ⌊ ℓ 2 ⌋ . Xujun Liu (UIUC) Long cycle May 18, 2019 5 / 23
Previous results In this talk, we denote P n the path on n vertices and M n the matching with n edges. A matching M in G is connected if all edges of M are in the same component of G . � 3 n − 2 � Gerencs´ er and Gy´ arf´ as (1967): R 2 ( P n ) = . They actually 2 showed for positive integers k ≥ ℓ , R ( P k , P ℓ ) = k − 1 + ⌊ ℓ 2 ⌋ . Gy´ arf´ as, Ruszink´ o, S´ ark˝ ozy and Szemer´ edi (2007): � 2 n − 1 , if n is odd R 3 ( P n ) = 2 n − 2 , if n is even Xujun Liu (UIUC) Long cycle May 18, 2019 5 / 23
Previous results In this talk, we denote P n the path on n vertices and M n the matching with n edges. A matching M in G is connected if all edges of M are in the same component of G . � 3 n − 2 � Gerencs´ er and Gy´ arf´ as (1967): R 2 ( P n ) = . They actually 2 showed for positive integers k ≥ ℓ , R ( P k , P ℓ ) = k − 1 + ⌊ ℓ 2 ⌋ . Gy´ arf´ as, Ruszink´ o, S´ ark˝ ozy and Szemer´ edi (2007): � 2 n − 1 , if n is odd R 3 ( P n ) = 2 n − 2 , if n is even More colors? (For k ≥ 4) Xujun Liu (UIUC) Long cycle May 18, 2019 5 / 23
Previous results More colors (For k ≥ 4). Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
Previous results More colors (For k ≥ 4). Erd˝ os-Gallai (1959): Let G be a graph on n vertices. If e ( G ) > ℓ ( n − 1) / 2, then G contains a cycle of length at least ℓ + 1. It implies R k ( P n ) ≤ kn . Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
Previous results More colors (For k ≥ 4). Erd˝ os-Gallai (1959): Let G be a graph on n vertices. If e ( G ) > ℓ ( n − 1) / 2, then G contains a cycle of length at least ℓ + 1. It implies R k ( P n ) ≤ kn . k S´ ark˝ ozy (2016): R k ( P n ) ≤ ( k − 16 k 3 +1 ) n . Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
Previous results More colors (For k ≥ 4). Erd˝ os-Gallai (1959): Let G be a graph on n vertices. If e ( G ) > ℓ ( n − 1) / 2, then G contains a cycle of length at least ℓ + 1. It implies R k ( P n ) ≤ kn . k S´ ark˝ ozy (2016): R k ( P n ) ≤ ( k − 16 k 3 +1 ) n . Davis, Jenssen and Roberts (2017): R k ( P n ) ≤ ( k − 1 4 + 1 2 k ) n . Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
Previous results More colors (For k ≥ 4). Erd˝ os-Gallai (1959): Let G be a graph on n vertices. If e ( G ) > ℓ ( n − 1) / 2, then G contains a cycle of length at least ℓ + 1. It implies R k ( P n ) ≤ kn . k S´ ark˝ ozy (2016): R k ( P n ) ≤ ( k − 16 k 3 +1 ) n . Davis, Jenssen and Roberts (2017): R k ( P n ) ≤ ( k − 1 4 + 1 2 k ) n . Knierim and Su (2019): R k ( C n ) ≤ ( k − 1 2 + o (1)) n , where n is even. It implies R k ( P n ) ≤ ( k − 1 2 + o (1)) n . Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
Previous results More colors (For k ≥ 4). Erd˝ os-Gallai (1959): Let G be a graph on n vertices. If e ( G ) > ℓ ( n − 1) / 2, then G contains a cycle of length at least ℓ + 1. It implies R k ( P n ) ≤ kn . k S´ ark˝ ozy (2016): R k ( P n ) ≤ ( k − 16 k 3 +1 ) n . Davis, Jenssen and Roberts (2017): R k ( P n ) ≤ ( k − 1 4 + 1 2 k ) n . Knierim and Su (2019): R k ( C n ) ≤ ( k − 1 2 + o (1)) n , where n is even. It implies R k ( P n ) ≤ ( k − 1 2 + o (1)) n . Sun, Yang, Xu and Li (2006): R k ( P n ) ≥ ( k − 1 + o (1)) n . Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
Previous results Figaj and � Luczak used the Szemer´ edi regularity lemma to show that in the case of Ramsey numbers asymptotically, avoiding connected matchings, paths and even cycles are the same. Lemma Let a real number c > 0 and a positive integer k be given. If for every ǫ > 0 there exists a δ > 0 and an n 0 such that for every even n > n 0 and � v ( G ) � each graph G with v ( G ) > (1 + ǫ ) cn and e ( G ) ≥ (1 − δ ) and each 2 k-edge-coloring of G has a monochromatic connected matching M n / 2 , then for sufficiently large n, R k ( C n ) ≤ ( c + o (1)) n. Hence, we also have R k ( P n ) ≤ ( c + o (1)) n. Xujun Liu (UIUC) Long cycle May 18, 2019 7 / 23
Host graph: complete bipartite graph Gy´ arf´ as and Lehel (1973), Faudree and Schelp (1975): For positive integers n , K n , n �→ ( P 2 ⌈ n / 2 ⌉ , P 2 ⌈ n / 2 ⌉ ). Furthermore, K n , n �→ ( P 2 ⌈ n / 2 ⌉ +1 , P 2 ⌈ n / 2 ⌉ +1 ). Xujun Liu (UIUC) Long cycle May 18, 2019 8 / 23
Host graph: complete bipartite graph Gy´ arf´ as and Lehel (1973), Faudree and Schelp (1975): For positive integers n , K n , n �→ ( P 2 ⌈ n / 2 ⌉ , P 2 ⌈ n / 2 ⌉ ). Furthermore, K n , n �→ ( P 2 ⌈ n / 2 ⌉ +1 , P 2 ⌈ n / 2 ⌉ +1 ). | V ′ 1 | = n | V ′′ 1 | = n | V 1 | = 2 n | V 2 | = 2 n | V ′ 2 | = n | V ′′ 2 | = n Xujun Liu (UIUC) Long cycle May 18, 2019 8 / 23
Host graph: complete bipartite graph DeBiasio and Krueger (2019+): Let n be a sufficiently large positive integer. Then K n , n �→ ( C ≥ 2 ⌊ n / 2 ⌋ , C ≥ 2 ⌊ n / 2 ⌋ ). Moreover, they also showed For all 0 < δ < 1 and ǫ > 0, there exists n 0 such that if G is a balanced bipartite graph with 2 n ≥ 2 n 0 vertices and δ ( G ) ≥ δ n , then G �→ ( C ≥ ( f ( δ ) − ǫ ) n , C ≥ ( f ( δ ) − ǫ ) n ) , where δ, 0 ≤ δ ≤ 2 / 3 f ( δ ) = 4 δ − 2 , 2 / 3 ≤ δ ≤ 3 / 4 1 , 3 / 4 ≤ δ ≤ 1 . Xujun Liu (UIUC) Long cycle May 18, 2019 9 / 23
Host graph: complete bipartite graph DeBiasio and Krueger (2019+): Let n be a sufficiently large positive integer. Then K n , n �→ ( C ≥ 2 ⌊ n / 2 ⌋ , C ≥ 2 ⌊ n / 2 ⌋ ). Moreover, they also showed For all 0 < δ < 1 and ǫ > 0, there exists n 0 such that if G is a balanced bipartite graph with 2 n ≥ 2 n 0 vertices and δ ( G ) ≥ δ n , then G �→ ( C ≥ ( f ( δ ) − ǫ ) n , C ≥ ( f ( δ ) − ǫ ) n ) , where δ, 0 ≤ δ ≤ 2 / 3 f ( δ ) = 4 δ − 2 , 2 / 3 ≤ δ ≤ 3 / 4 1 , 3 / 4 ≤ δ ≤ 1 . Bucic, Letzter and Sudakov (2018): For positive integers n , K n , n �→ ( P 2 n 3 − o ( n ) , P 2 n 3 − o ( n ) , P 2 n 3 − o ( n ) ). Xujun Liu (UIUC) Long cycle May 18, 2019 9 / 23
Host graph: complete tripartite graph Gy´ arf´ as, Ruszink´ o, S´ ark˝ ozy and Szemer´ edi (2007): For positive integers n , K n , n , n �→ ( P 2 n − o ( n ) , P 2 n − o ( n ) ). Xujun Liu (UIUC) Long cycle May 18, 2019 10 / 23
Host graph: complete tripartite graph Gy´ arf´ as, Ruszink´ o, S´ ark˝ ozy and Szemer´ edi (2007): For positive integers n , K n , n , n �→ ( P 2 n − o ( n ) , P 2 n − o ( n ) ). Conjecture (Gy´ arf´ as, Ruszink´ o, S´ ark˝ ozy and Szemer´ edi (2007)) K n , n , n �→ ( P 2 n +1 , P 2 n +1 ) . Xujun Liu (UIUC) Long cycle May 18, 2019 10 / 23
Host graph: complete tripartite graph Gy´ arf´ as, Ruszink´ o, S´ ark˝ ozy and Szemer´ edi (2007): For positive integers n , K n , n , n �→ ( P 2 n − o ( n ) , P 2 n − o ( n ) ). Conjecture (Gy´ arf´ as, Ruszink´ o, S´ ark˝ ozy and Szemer´ edi (2007)) K n , n , n �→ ( P 2 n +1 , P 2 n +1 ) . Theorem (Balogh, Kostochka, Lavrov, L. (2019+)) Let n be sufficiently large. Then K n , n , n �→ ( P 2 n +1 , P 2 n +1 ) . Xujun Liu (UIUC) Long cycle May 18, 2019 10 / 23
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