Long monochromatic paths and cycles in 2-edge-colored multipartite - - PowerPoint PPT Presentation
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,
Xujun Liu
University of Illinois at Urbana-Champaign Joint work with J´
31st Cumberland Conference on Combinatorics, Graph Theory and Computing
May 18, 2019
Xujun Liu (UIUC) Long cycle May 18, 2019 1 / 23
1
Introduction
2
Main results
Xujun Liu (UIUC) Long cycle May 18, 2019 2 / 23
1
Introduction
2
Main results
Xujun Liu (UIUC) Long cycle May 18, 2019 3 / 23
For graphs G0, . . . , Gk we write G0 → (G1, . . . , Gk) if for every k-coloring
edges of color i. The Ramsey number Rk(G) is the minimum N such that KN → (G1, . . . , Gk), where G1 = . . . = Gk = G.
Xujun Liu (UIUC) Long cycle May 18, 2019 4 / 23
For graphs G0, . . . , Gk we write G0 → (G1, . . . , Gk) if for every k-coloring
edges of color i. The Ramsey number Rk(G) is the minimum N such that KN → (G1, . . . , Gk), where G1 = . . . = Gk = G.
Example
R2(K3) = 6.
Figure: R2(K3) = 6.
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In this talk, we denote Pn the path on n vertices and Mn 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
In this talk, we denote Pn the path on n vertices and Mn the matching with n edges. A matching M in G is connected if all edges of M are in the same component of G. Gerencs´ er and Gy´ arf´ as (1967): R2(Pn) = 3n−2
2
showed for positive integers k ≥ ℓ, R(Pk, Pℓ) = k − 1 + ⌊ ℓ
2⌋.
Xujun Liu (UIUC) Long cycle May 18, 2019 5 / 23
In this talk, we denote Pn the path on n vertices and Mn the matching with n edges. A matching M in G is connected if all edges of M are in the same component of G. Gerencs´ er and Gy´ arf´ as (1967): R2(Pn) = 3n−2
2
showed for positive integers k ≥ ℓ, R(Pk, Pℓ) = k − 1 + ⌊ ℓ
2⌋.
Gy´ arf´ as, Ruszink´
ark˝
edi (2007): R3(Pn) =
if n is odd 2n − 2, if n is even
Xujun Liu (UIUC) Long cycle May 18, 2019 5 / 23
In this talk, we denote Pn the path on n vertices and Mn the matching with n edges. A matching M in G is connected if all edges of M are in the same component of G. Gerencs´ er and Gy´ arf´ as (1967): R2(Pn) = 3n−2
2
showed for positive integers k ≥ ℓ, R(Pk, Pℓ) = k − 1 + ⌊ ℓ
2⌋.
Gy´ arf´ as, Ruszink´
ark˝
edi (2007): R3(Pn) =
if n is odd 2n − 2, if n is even More colors? (For k ≥ 4)
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More colors (For k ≥ 4).
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More colors (For k ≥ 4). Erd˝
e(G) > ℓ(n − 1)/2, then G contains a cycle of length at least ℓ + 1. It implies Rk(Pn) ≤ kn.
Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
More colors (For k ≥ 4). Erd˝
e(G) > ℓ(n − 1)/2, then G contains a cycle of length at least ℓ + 1. It implies Rk(Pn) ≤ kn. S´ ark˝
k 16k3+1)n.
Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
More colors (For k ≥ 4). Erd˝
e(G) > ℓ(n − 1)/2, then G contains a cycle of length at least ℓ + 1. It implies Rk(Pn) ≤ kn. S´ ark˝
k 16k3+1)n.
Davis, Jenssen and Roberts (2017): Rk(Pn) ≤ (k − 1
4 + 1 2k )n.
Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
More colors (For k ≥ 4). Erd˝
e(G) > ℓ(n − 1)/2, then G contains a cycle of length at least ℓ + 1. It implies Rk(Pn) ≤ kn. S´ ark˝
k 16k3+1)n.
Davis, Jenssen and Roberts (2017): Rk(Pn) ≤ (k − 1
4 + 1 2k )n.
Knierim and Su (2019): Rk(Cn) ≤ (k − 1
2 + o(1))n, where n is even. It
implies Rk(Pn) ≤ (k − 1 2 + o(1))n.
Xujun Liu (UIUC) Long cycle May 18, 2019 6 / 23
More colors (For k ≥ 4). Erd˝
e(G) > ℓ(n − 1)/2, then G contains a cycle of length at least ℓ + 1. It implies Rk(Pn) ≤ kn. S´ ark˝
k 16k3+1)n.
Davis, Jenssen and Roberts (2017): Rk(Pn) ≤ (k − 1
4 + 1 2k )n.
Knierim and Su (2019): Rk(Cn) ≤ (k − 1
2 + o(1))n, where n is even. It
implies Rk(Pn) ≤ (k − 1 2 + o(1))n. Sun, Yang, Xu and Li (2006): Rk(Pn) ≥ (k − 1 + o(1))n.
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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 n0 such that for every even n > n0 and each graph G with v(G) > (1 + ǫ)cn and e(G) ≥ (1 − δ) v(G)
2
k-edge-coloring of G has a monochromatic connected matching Mn/2, then for sufficiently large n, Rk(Cn) ≤ (c + o(1))n. Hence, we also have Rk(Pn) ≤ (c + o(1))n.
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Gy´ arf´ as and Lehel (1973), Faudree and Schelp (1975): For positive integers n, Kn,n → (P2⌈n/2⌉, P2⌈n/2⌉). Furthermore, Kn,n → (P2⌈n/2⌉+1, P2⌈n/2⌉+1).
Xujun Liu (UIUC) Long cycle May 18, 2019 8 / 23
Gy´ arf´ as and Lehel (1973), Faudree and Schelp (1975): For positive integers n, Kn,n → (P2⌈n/2⌉, P2⌈n/2⌉). Furthermore, Kn,n → (P2⌈n/2⌉+1, P2⌈n/2⌉+1).
|V1| = 2n |V2| = 2n
|V ′′
2 | = n|V ′
2| = n|V ′′
1 | = n|V ′
1| = nXujun Liu (UIUC) Long cycle May 18, 2019 8 / 23
DeBiasio and Krueger (2019+): Let n be a sufficiently large positive
For all 0 < δ < 1 and ǫ > 0, there exists n0 such that if G is a balanced bipartite graph with 2n ≥ 2n0 vertices and δ(G) ≥ δn, then G → (C≥(f (δ)−ǫ)n, C≥(f (δ)−ǫ)n), where f (δ) = δ, 0 ≤ δ ≤ 2/3 4δ − 2, 2/3 ≤ δ ≤ 3/4 1, 3/4 ≤ δ ≤ 1.
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DeBiasio and Krueger (2019+): Let n be a sufficiently large positive
For all 0 < δ < 1 and ǫ > 0, there exists n0 such that if G is a balanced bipartite graph with 2n ≥ 2n0 vertices and δ(G) ≥ δn, then G → (C≥(f (δ)−ǫ)n, C≥(f (δ)−ǫ)n), where f (δ) = δ, 0 ≤ δ ≤ 2/3 4δ − 2, 2/3 ≤ δ ≤ 3/4 1, 3/4 ≤ δ ≤ 1. Bucic, Letzter and Sudakov (2018): For positive integers n, Kn,n → (P 2n
3 −o(n), P 2n 3 −o(n), P 2n 3 −o(n)). Xujun Liu (UIUC) Long cycle May 18, 2019 9 / 23
Gy´ arf´ as, Ruszink´
ark˝
edi (2007): For positive integers n, Kn,n,n → (P2n−o(n), P2n−o(n)).
Xujun Liu (UIUC) Long cycle May 18, 2019 10 / 23
Gy´ arf´ as, Ruszink´
ark˝
edi (2007): For positive integers n, Kn,n,n → (P2n−o(n), P2n−o(n)).
Conjecture (Gy´ arf´ as, Ruszink´
ark˝
edi (2007))
Kn,n,n → (P2n+1, P2n+1).
Xujun Liu (UIUC) Long cycle May 18, 2019 10 / 23
Gy´ arf´ as, Ruszink´
ark˝
edi (2007): For positive integers n, Kn,n,n → (P2n−o(n), P2n−o(n)).
Conjecture (Gy´ arf´ as, Ruszink´
ark˝
edi (2007))
Kn,n,n → (P2n+1, P2n+1).
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let n be sufficiently large. Then Kn,n,n → (P2n+1, P2n+1).
Xujun Liu (UIUC) Long cycle May 18, 2019 10 / 23
Gy´ arf´ as, Ruszink´
ark˝
edi (2007): For positive integers n, Kn,n,n → (P2n−o(n), P2n−o(n)).
Conjecture (Gy´ arf´ as, Ruszink´
ark˝
edi (2007))
Kn,n,n → (P2n+1, P2n+1).
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let n be sufficiently large. Then Kn,n,n → (P2n+1, P2n+1).
|U1| = 2n |U2| = n − 1
Xujun Liu (UIUC) Long cycle May 18, 2019 10 / 23
1
Introduction
2
Main results
Xujun Liu (UIUC) Long cycle May 18, 2019 11 / 23
We consider Kn1,...,ns with n1 ≥ n2 ≥ . . . ≥ ns ≥ 1.
Question
Under what conditions does every 2-edge-coloring of Kn1,...,ns contains a monochromatic a) P2n; b) P2n+1; c) C2n; d) C≥2n?
Xujun Liu (UIUC) Long cycle May 18, 2019 12 / 23
We consider Kn1,...,ns with n1 ≥ n2 ≥ . . . ≥ ns ≥ 1.
Question
Under what conditions does every 2-edge-coloring of Kn1,...,ns contains a monochromatic a) P2n; b) P2n+1; c) C2n; d) C≥2n?
Remark
A connected matching Mn is contained in each of P2n, P2n+1, C2n and C≥2n.
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Condition 1: N = n1 + . . . + ns ≥ 3n − 1. (1)
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Condition 1: N = n1 + . . . + ns ≥ 3n − 1. (1)
Example
No mono Mn and hence no mono P2n, P2n+1, C≥2n.
|U1| = 2n − 1 |U2| = n − 1
Figure: Condition 1 is necessary.
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Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. (2)
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Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. (2)
Example
No mono Mn and hence no mono P2n, P2n+1, C≥2n.
|U1| = n1 |U2| = n − 1 |U3| = n − 1
Figure: Condition 2 is necessary.
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Conditions 1 and 2 are together sufficient for Kn1,...,ns → (Mn, Mn).
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let s ≥ 2 and G be a complete s-partite graph Kn1,...,ns such that N := n1 + . . . + ns ≥ 3n − 1, and N − ni ≥ 2n − 1 for every 1 ≤ i ≤ s. Let E(G) = E1 ∪ E2 be a partition of the edges of G, and let Gi = G[Ei] for i = 1, 2. Then for some i, α′
∗(Gi) ≥ n, i.e., Gi contains Mn.
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Condition 1: N = n1 + . . . + ns ≥ 3n − 1. Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1.
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Condition 1: N = n1 + . . . + ns ≥ 3n − 1. Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Conditions 1 and 2 are together also sufficient for Kn1,...,ns → (P2n, P2n).
Xujun Liu (UIUC) Long cycle May 18, 2019 16 / 23
Condition 1: N = n1 + . . . + ns ≥ 3n − 1. Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Conditions 1 and 2 are together also sufficient for Kn1,...,ns → (P2n, P2n).
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let s ≥ 2 and n be sufficiently large. Let n1 ≥ . . . ≥ ns and N = n1 + . . . + ns satisfy Conditions 1 and 2. Then for each 2-edge-coloring f of the edges of the complete s-partite graph Kn1,...,ns, there exists a monochromatic path P2n.
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For P2n+1, except Condition 2, we need Condition 3: N ≥ 3n. (3)
Example
No red Mn and no blue P2n+1.
|U1| = 2n |U2| = n − 1
Figure: Condition 3 is necessary.
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For P2n+1, except Conditions 2 and 3, we need Condition 4: if n3 = 0 then n1 ≥ 2n + 1. (4)
Example
No monochromatic P2n+1.
|V1| = 2n |V2| = 2n
|V ′′
2 | = n|V ′
2| = n|V ′′
1 | = n|V ′
1| = nFigure: Condition 4 is necessary.
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Conditions 2,3,4 together are sufficient for Kn1,...,ns → (P2n+1, P2n+1).
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Conditions 2,3,4 together are sufficient for Kn1,...,ns → (P2n+1, P2n+1). Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Condition 3: N ≥ 3n. Condition 4: if n3 = 0 then n1 ≥ 2n + 1.
Xujun Liu (UIUC) Long cycle May 18, 2019 19 / 23
Conditions 2,3,4 together are sufficient for Kn1,...,ns → (P2n+1, P2n+1). Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Condition 3: N ≥ 3n. Condition 4: if n3 = 0 then n1 ≥ 2n + 1.
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let s ≥ 2 and n be sufficiently large. Let n1 ≥ . . . ≥ ns and N = n1 + . . . + ns satisfy Condition 2,3 and 4. Then for each 2-edge-coloring f of the edges of the complete s-partite graph Kn1,...,ns, there exists a monochromatic path P2n+1.
Xujun Liu (UIUC) Long cycle May 18, 2019 19 / 23
Conditions 2,3,4 together are sufficient for Kn1,...,ns → (P2n+1, P2n+1). Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Condition 3: N ≥ 3n. Condition 4: if n3 = 0 then n1 ≥ 2n + 1.
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let s ≥ 2 and n be sufficiently large. Let n1 ≥ . . . ≥ ns and N = n1 + . . . + ns satisfy Condition 2,3 and 4. Then for each 2-edge-coloring f of the edges of the complete s-partite graph Kn1,...,ns, there exists a monochromatic path P2n+1.
Corollary
Let n be sufficiently large. Then Kn,n,n → (P2n+1, P2n+1).
Xujun Liu (UIUC) Long cycle May 18, 2019 19 / 23
Condition 1: N = n1 + . . . + ns ≥ 3n − 1. Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Condition 5: if N − n1 − n2 ≤ 2, then n1 ≥ 2n − 1. Condition 6: if N − n1 − n2 ≤ 1, then n1 + N ≥ 6n − 2.
Xujun Liu (UIUC) Long cycle May 18, 2019 20 / 23
Condition 1: N = n1 + . . . + ns ≥ 3n − 1. Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Condition 5: if N − n1 − n2 ≤ 2, then n1 ≥ 2n − 1. Condition 6: if N − n1 − n2 ≤ 1, then n1 + N ≥ 6n − 2.
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let s ≥ 2 and n be sufficiently large. Let n1 ≥ . . . ≥ ns and N = n1 + . . . + ns satisfy Condition 1,2,5 and 6. Then for each 2-edge-coloring f of the edges of the complete s-partite graph Kn1,...,ns, there exists a monochromatic cycle C≥2n.
Xujun Liu (UIUC) Long cycle May 18, 2019 20 / 23
Condition 1: N = n1 + . . . + ns ≥ 3n − 1. Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Condition 7: if N − n1 − n2 ≤ 2, then N ≥ 4n − 1.
Xujun Liu (UIUC) Long cycle May 18, 2019 21 / 23
Condition 1: N = n1 + . . . + ns ≥ 3n − 1. Condition 2: N − n1 = n2 + . . . + ns ≥ 2n − 1. Condition 7: if N − n1 − n2 ≤ 2, then N ≥ 4n − 1.
Theorem (Balogh, Kostochka, Lavrov, L. (2019+))
Let s ≥ 2 and n be sufficiently large. Let n1 ≥ . . . ≥ ns and N = n1 + . . . + ns satisfy Condition 1,2 and 7. Then for each 2-edge-coloring f of the edges of the complete s-partite graph Kn1,...,ns, there exists a monochromatic cycle C2n.
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Possible work: Improve the following result.
Xujun Liu (UIUC) Long cycle May 18, 2019 22 / 23
Possible work: Improve the following result.
Theorem (Knierim and Su (2019))
Rk(Pn) ≤ (k − 1
2 + o(1))n.
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Xujun Liu (UIUC) Long cycle May 18, 2019 23 / 23