Polychromatic Colorings of Complete Graphs with Respect to - PowerPoint PPT Presentation

Polychromatic Colorings of Complete Graphs with Respect to 1-,2-factors and Hamiltonian Cycles Maria Axenovich John Goldwasser Ryan Hansen Bernard Lidick´ y Ryan R. Martin David Offner John Talbot Michael Young SIAM DM June 6, 2018

Polychromatic Coloring Let G and H be graphs and C a set of colors. Let ϕ : E ( G ) → C (not necessarily proper edge-coloring) ϕ is an H-polychromatic coloring of G if every subgraph of G isomorphic to H contains all colors in C . Example H = K 3 , G = K 4 , C = { red , blue } . 2

Polychromatic Coloring Let G and H be graphs and C a set of colors. Let ϕ : E ( G ) → C (not necessarily proper edge-coloring) ϕ is an H-polychromatic coloring of G if every subgraph of G isomorphic to H contains all colors in C . Example H = K 3 , G = K 4 , C = { red , blue } . 2

Polychromatic Coloring Let G and H be graphs and C a set of colors. Let ϕ : E ( G ) → C (not necessarily proper edge-coloring) ϕ is an H-polychromatic coloring of G if every subgraph of G isomorphic to H contains all colors in C . Example H = K 3 , G = K 4 , C = { red , blue } . 2

H -polychromatic Number ϕ is a H-polychromatic coloring of G with respect to H if every subgraph of G isomorphic to H contains all colors in C . Easier to find ϕ with fewer colors. 3

H -polychromatic Number ϕ is a H-polychromatic coloring of G with respect to H if every subgraph of G isomorphic to H contains all colors in C . Easier to find ϕ with fewer colors. 3

H -polychromatic Number ϕ is a H-polychromatic coloring of G with respect to H if every subgraph of G isomorphic to H contains all colors in C . Easier to find ϕ with fewer colors. H-polychromatic number of G is the maximum number of colors k such that there exists a polychromatic coloring of G with respect to H using k colors. Notation poly H ( G ) = k Example poly K 3 ( K 4 ) = 3 3

Motivation for H -polychromatic Number Let Q d be a d -dimensional hypercube. Problem What is the lergest X ⊆ E ( Q n ) such that Q n [ X ] is Q d -free? ex ( Q n , Q d )? Example for Q 2 in Q 3 . 4

Motivation for H -polychromatic Number Let Q d be a d -dimensional hypercube. Problem What is the lergest X ⊆ E ( Q n ) such that Q n [ X ] is Q d -free? ex ( Q n , Q d )? Example for Q 2 in Q 3 . 4

Motivation for H -polychromatic Number Let Q d be a d -dimensional hypercube. Problem What is the lergest X ⊆ E ( Q n ) such that Q n [ X ] is Q d -free? ex ( Q n , Q d )? Example for Q 2 in Q 3 . Any color class of any Q d -polychromatic coloring of Q n gives a lower bound on | X | . e ( Q n )(1 − 1 / poly Q d ( Q n )) ≤ ex( Q n , Q d ) 4

Motivation for H -polychromatic Number Let Q d be a d -dimensional hypercube. Problem What is the lergest X ⊆ E ( Q n ) such that Q n [ X ] is Q d -free? ex ( Q n , Q d )? Example for Q 2 in Q 3 . Any color class of any Q d -polychromatic coloring of Q n gives a lower bound on | X | . e ( Q n )(1 − 1 / poly Q d ( Q n )) ≤ ex( Q n , Q d ) 4

Known Results Theorem (Alon, Krech, Szab´ o 2007) � ( d +1) 2 � d + 1 � if d is odd 4 ≥ poly Q d ( Q n ) ≥ d ( d +2) 2 if d is even 4 Theorem (Offner 2008) � ( d +1) 2 if d is odd 4 poly Q d ( Q n ) = d ( d +2) if d is even 4 5

Anti-Ramsey Edge coloring of H is rainbow if no two edges of H receive the same color. Edge coloring of G is H-anti-ramsey if NO copy of H in G is rainbow. ar ( G , H ) is the largest number of colors used in an H -anti-Ramsey coloring of G . ar ( G , H ) ≤ ex ( G , H ) � 2 � ar ( G , H ) ≥ 1 − e ( G ) poly H ( G ) 6

Polychromatic Coloring of Integers Let S ⊂ Z be finite. Coloring of Z is S-polychromatic if every translation of S contains all colors. Example: S = { 0 , 1 , 4 , 5 } 1 1 1 1 2 3 2 3 1 1 1 All about this during the next talk in this session by John Goldwasser. 7

Polychromatic Coloring of Integers Let S ⊂ Z be finite. Coloring of Z is S-polychromatic if every translation of S contains all colors. Example: S = { 0 , 1 , 4 , 5 } 1 1 1 1 1 1 2 2 3 3 2 3 1 1 1 All about this during the next talk in this session by John Goldwasser. 7

Polychromatic Coloring of Integers Let S ⊂ Z be finite. Coloring of Z is S-polychromatic if every translation of S contains all colors. Example: S = { 0 , 1 , 4 , 5 } 1 1 1 1 1 1 2 3 3 2 2 3 1 1 1 All about this during the next talk in this session by John Goldwasser. 7

Polychromatic Coloring of Integers Let S ⊂ Z be finite. Coloring of Z is S-polychromatic if every translation of S contains all colors. Example: S = { 0 , 1 , 4 , 5 } 1 1 1 1 1 1 2 3 2 2 3 3 1 1 1 All about this during the next talk in this session by John Goldwasser. 7

Polychromatic Coloring of Integers Let S ⊂ Z be finite. Coloring of Z is S-polychromatic if every translation of S contains all colors. Example: S = { 0 , 1 , 4 , 5 } 1 1 1 1 1 2 2 3 2 3 3 1 1 1 1 All about this during the next talk in this session by John Goldwasser. 7

Polychromatic Coloring of Integers Let S ⊂ Z be finite. Coloring of Z is S-polychromatic if every translation of S contains all colors. Example: S = { 0 , 1 , 4 , 5 } 1 1 1 1 2 2 3 3 2 3 1 1 1 1 1 All about this during the next talk in this session by John Goldwasser. 7

Polychromatic Coloring of Integers Let S ⊂ Z be finite. Coloring of Z is S-polychromatic if every translation of S contains all colors. Example: S = { 0 , 1 , 4 , 5 } 1 1 1 1 2 3 3 2 2 3 1 1 1 1 1 All about this during the next talk in this session by John Goldwasser. 7

Our Results for This Talk Let F k be a k -factor and HC be a Hamiltonian Cycle. Theorem (AGHLMOTY ’18) If n is an even positive integer, then poly F 1 ( K n ) = ⌊ log 2 n ⌋ . Theorem (AGHLMOTY ’18) There exists a constant c such that ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) ≤ poly HC ( K n ) ≤ log 2 n + c . Exact solution for poly F 2 ( K n ) and poly HC ( K n ) by G&H. 8

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Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Constructions For Lower Bounds ⌊ log 2 n ⌋ ≤ poly F 1 ( K n ) ⌊ log 2 2( n + 1) ⌋ ≤ poly F 2 ( K n ) 10

Upper bound for poly F 1 ( K n ) • show there is an optimal coloring that has ordering of vertices such that for each fixed vertex v “all edges going to the right have the same color”. • for ever vertex define inherited color , counting argument using majority. 11

Upper bound for poly F 1 ( K n ) • show there is an optimal coloring that has ordering of vertices such that for each fixed vertex v “all edges going to the right have the same color”. • for ever vertex define inherited color , counting argument using majority. 11

Counting first for poly F 1 ( K n ) for vertex define inherited color 12

Counting first for poly F 1 ( K n ) for vertex define inherited color Let M c be vertices colored color c ∈ { 1 , 2 , . . . } . Feature: ∀ c exists i c ∈ [ n ] such that | M c ∩ { v 1 , . . . , v i }| > i / 2. 12

Counting first for poly F 1 ( K n ) for vertex define inherited color Let M c be vertices colored color c ∈ { 1 , 2 , . . . } . Feature: ∀ c exists i c ∈ [ n ] such that | M c ∩ { v 1 , . . . , v i }| > i / 2. 12

Counting first for poly F 1 ( K n ) for vertex define inherited color Let M c be vertices colored color c ∈ { 1 , 2 , . . . } . Feature: ∀ c exists i c ∈ [ n ] such that | M c ∩ { v 1 , . . . , v i }| > i / 2. 12