Rainbow Hamilton cycles Po-Shen Loh Carnegie Mellon University Joint work with Alan Frieze
Graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once.
Graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once. Definition Erd˝ os-R´ enyi G n , p : edges appear independently with probability p .
Graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once. Definition Erd˝ os-R´ enyi G n , p : edges appear independently with probability p . (Koml´ os, Szemer´ edi; Bollob´ as) G n , p is Hamiltonian whp if p = log n +log log n + ω ( n ) with ω ( n ) → ∞ . n
Graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once. Definition Erd˝ os-R´ enyi G n , p : edges appear independently with probability p . (Koml´ os, Szemer´ edi; Bollob´ as) G n , p is Hamiltonian whp if p = log n +log log n + ω ( n ) with ω ( n ) → ∞ . n (Robinson, Wormald) G 3-reg is Hamiltonian whp .
Graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once. Definition Erd˝ os-R´ enyi G n , p : edges appear independently with probability p . (Koml´ os, Szemer´ edi; Bollob´ as) G n , p is Hamiltonian whp if p = log n +log log n + ω ( n ) with ω ( n ) → ∞ . n (Robinson, Wormald) G 3-reg is Hamiltonian whp . (Bohman, Frieze) G 3-out is Hamiltonian whp .
Graphs Definition A cycle is Hamiltonian if it visits every vertex exactly once. Definition Erd˝ os-R´ enyi G n , p : edges appear independently with probability p . (Koml´ os, Szemer´ edi; Bollob´ as) G n , p is Hamiltonian whp if p = log n +log log n + ω ( n ) with ω ( n ) → ∞ . n (Robinson, Wormald) G 3-reg is Hamiltonian whp . (Bohman, Frieze) G 3-out is Hamiltonian whp . (Cooper, Frieze) D 2-in,2-out is Hamiltonian whp .
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p .
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p .
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p .
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p .
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p .
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle (Frieze) H n , p ;3 has loose H-cycle whp if p > K log n , 4 | n . n 2
Hypergraphs Definition (3-uniform hypergraph) H n , p ;3 : each triple appears independently with probability p . Tight H-cycle Loose H-cycle (Frieze) H n , p ;3 has loose H-cycle whp if p > K log n , 4 | n . n 2 (Dudek, Frieze) Asymptotically answered for all uniformities, and all degrees of loose-ness.
Rainbow Hamilton cycles Question Does G n , p have a rainbow Hamilton cycle if edges are randomly colored from κ colors?
Rainbow Hamilton cycles Question Does G n , p have a rainbow Hamilton cycle if edges are randomly colored from κ colors? Observations Must have p > log n +log log n + ω ( n ) with ω ( n ) → ∞ . n Must have κ ≥ n .
Rainbow Hamilton cycles Question Does G n , p have a rainbow Hamilton cycle if edges are randomly colored from κ colors? Observations Must have p > log n +log log n + ω ( n ) with ω ( n ) → ∞ . n Must have κ ≥ n . (Cooper, Frieze) True if p = 20 log n and κ = 20 n . n
Rainbow Hamilton cycles Question Does G n , p have a rainbow Hamilton cycle if edges are randomly colored from κ colors? Observations Must have p > log n +log log n + ω ( n ) with ω ( n ) → ∞ . n Must have κ ≥ n . (Cooper, Frieze) True if p = 20 log n and κ = 20 n . n (Janson, Wormald) True if G 2 r -reg is randomly colored with each of κ = n colors appearing exactly r ≥ 4 times.
Hypergraph (bisected vertex set) Loose vs. rainbow H-cycles Connect 3-uniform hypergraphs to colored graphs
Loose vs. rainbow H-cycles Connect 3-uniform hypergraphs to colored graphs. Hypergraph (bisected vertex set)
Loose vs. rainbow H-cycles Connect 3-uniform hypergraphs to colored graphs. Hypergraph (bisected vertex set)
Loose vs. rainbow H-cycles Connect 3-uniform hypergraphs to colored graphs. Hypergraph Auxiliary graph (bisected vertex set)
Loose vs. rainbow H-cycles Connect 3-uniform hypergraphs (loose Hamiltonicity) to colored graphs (rainbow Hamilton cycles). Hypergraph Auxiliary graph (bisected vertex set)
Loose vs. rainbow H-cycles Connect 3-uniform hypergraphs (loose Hamiltonicity) to colored graphs (rainbow Hamilton cycles). Hypergraph Auxiliary graph (bisected vertex set) Frieze applied Johansson-Kahn-Vu to find perfect matchings. Apply Janson-Wormald to find rainbow H-cycle in randomly colored random regular graph.
Rainbow Hamilton cycles Theorem (Frieze, L.) For any fixed ǫ > 0, if p = (1+ ǫ ) log n , then G n , p contains a rainbow n Hamilton cycle whp when its edges are randomly colored from κ = (1 + ǫ ) n colors. Remarks: Asymptotically best possible, both in terms of p and κ . Still holds when ǫ tends (slowly) to zero.
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n Justification: Degree of fixed vertex is Bin [ n − 1 , p ]; expectation ∼ log n
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n Justification: Degree of fixed vertex is Bin [ n − 1 , p ]; expectation ∼ log n < e − 2 3 E = n − 2 deg( v ) < 1 � � 3 . By Chernoff, P 10 E
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n Justification: Degree of fixed vertex is Bin [ n − 1 , p ]; expectation ∼ log n < e − 2 3 E = n − 2 deg( v ) < 1 � � 3 . By Chernoff, P 10 E √ n vertices have degree ≥ 1 Typically, all but < 3 10 log n . �
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n First attempt to find rainbow H-cycle: Suppose all degrees ≥ 1 10 log n . At each vertex, expose list of colors that appear.
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n First attempt to find rainbow H-cycle: Suppose all degrees ≥ 1 10 log n . At each vertex, expose list of colors that appear. Select 3 colors per vertex s.t. all selected colors are different.
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n First attempt to find rainbow H-cycle: Suppose all degrees ≥ 1 10 log n . At each vertex, expose list of colors that appear. Select 3 colors per vertex s.t. all selected colors are different.
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n First attempt to find rainbow H-cycle: Suppose all degrees ≥ 1 10 log n . At each vertex, expose list of colors that appear. Select 3 colors per vertex s.t. all selected colors are different. Expose those edges only; like G 3-out .
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n First attempt to find rainbow H-cycle: Suppose all degrees ≥ 1 10 log n . At each vertex, expose list of colors that appear. Select 3 colors per vertex s.t. all selected colors are different. Expose those edges only; like G 3-out .
Proof ideas Observation If p = (1+ ǫ ) log n , then almost all vertices have degree ≥ 1 10 log n . n First attempt to find rainbow H-cycle: Suppose all degrees ≥ 1 10 log n . At each vertex, expose list of colors that appear. Select 3 colors per vertex s.t. all selected colors are different. Expose those edges only; like G 3-out . Already requires 3n colors.
A B A B A B A B A B Saving the constant factor Sprinkling Reserve p ′ = ǫ and κ ′ = ǫ n 2 · log n 2 for 2nd phase. n
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