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

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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

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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 = K3, G = K4, C = {red, blue}.

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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 = K3, G = K4, C = {red, blue}.

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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 = K3, G = K4, C = {red, blue}.

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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.

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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.

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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 polyH(G) = k Example polyK3(K4) = 3

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Motivation for H-polychromatic Number

Let Qd be a d-dimensional hypercube.

Problem

What is the lergest X ⊆ E(Qn) such that Qn[X] is Qd-free? ex(Qn, Qd)? Example for Q2 in Q3.

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Motivation for H-polychromatic Number

Let Qd be a d-dimensional hypercube.

Problem

What is the lergest X ⊆ E(Qn) such that Qn[X] is Qd-free? ex(Qn, Qd)? Example for Q2 in Q3.

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Motivation for H-polychromatic Number

Let Qd be a d-dimensional hypercube.

Problem

What is the lergest X ⊆ E(Qn) such that Qn[X] is Qd-free? ex(Qn, Qd)? Example for Q2 in Q3. Any color class of any Qd-polychromatic coloring of Qn gives a lower bound on |X|. e(Qn)(1 − 1/polyQd(Qn)) ≤ ex(Qn, Qd)

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Motivation for H-polychromatic Number

Let Qd be a d-dimensional hypercube.

Problem

What is the lergest X ⊆ E(Qn) such that Qn[X] is Qd-free? ex(Qn, Qd)? Example for Q2 in Q3. Any color class of any Qd-polychromatic coloring of Qn gives a lower bound on |X|. e(Qn)(1 − 1/polyQd(Qn)) ≤ ex(Qn, Qd)

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Known Results

Theorem (Alon, Krech, Szab´

2007)

d + 1 2

≥ polyQd(Qn) ≥

(d+1)2

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if d is odd

d(d+2) 4

if d is even

Theorem (Offner 2008)

polyQd(Qn) = (d+1)2

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if d is odd

d(d+2) 4

if d is even

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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) ar(G, H) ≥

1 −

2 polyH(G)

e(G)

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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.

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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 1 1 2 3 All about this during the next talk in this session by John Goldwasser.

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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 1 1 3 2 All about this during the next talk in this session by John Goldwasser.

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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 1 1 2 3 All about this during the next talk in this session by John Goldwasser.

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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 1 2 3 1 All about this during the next talk in this session by John Goldwasser.

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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 2 3 1 1 All about this during the next talk in this session by John Goldwasser.

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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 3 2 1 1 All about this during the next talk in this session by John Goldwasser.

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Our Results for This Talk

Let Fk be a k-factor and HC be a Hamiltonian Cycle.

Theorem (AGHLMOTY ’18)

If n is an even positive integer, then polyF1(Kn) = ⌊log2 n⌋.

Theorem (AGHLMOTY ’18)

There exists a constant c such that ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn) ≤ polyHC(Kn) ≤ log2 n + c. Exact solution for polyF2(Kn) and polyHC(Kn) by G&H.

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Constructions For Lower Bounds

⌊log2 n⌋ ≤ polyF1(Kn) ⌊log2 2(n + 1)⌋ ≤ polyF2(Kn)

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Upper bound for polyF1(Kn)

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.

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Upper bound for polyF1(Kn)

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.

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Counting first for polyF1(Kn)

for vertex define inherited color

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Counting first for polyF1(Kn)

for vertex define inherited color Let Mc be vertices colored color c ∈ {1, 2, . . .}. Feature: ∀c exists ic ∈ [n] such that |Mc ∩ {v1, . . . , vi}| > i/2.

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Counting first for polyF1(Kn)

for vertex define inherited color Let Mc be vertices colored color c ∈ {1, 2, . . .}. Feature: ∀c exists ic ∈ [n] such that |Mc ∩ {v1, . . . , vi}| > i/2.

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Counting first for polyF1(Kn)

for vertex define inherited color Let Mc be vertices colored color c ∈ {1, 2, . . .}. Feature: ∀c exists ic ∈ [n] such that |Mc ∩ {v1, . . . , vi}| > i/2.

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Counting first for polyF1(Kn)

for vertex define inherited color Let Mc be vertices colored color c ∈ {1, 2, . . .}. Feature: ∀c exists ic ∈ [n] such that |Mc ∩ {v1, . . . , vi}| > i/2. Assume that i1 < i2 < . . .. By induction |Mc| ≥ 2c − 1.

c

|Mc| ≤ n = ⇒ c ≤ ⌊log2 n⌋

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Ordering the vertices

Take largest ordered initial segment,

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Ordering the vertices

v y1 y2 y3 y4

. . .

yd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest,

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Ordering the vertices

v u y1 y2 y3 y4

. . .

yd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv,

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Ordering the vertices

v u y1 y2 y3 y4

. . .

yd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue,

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment,

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment,

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment,

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment,

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment,

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment,

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment, u has higher mono degree than v.

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Ordering the vertices

v u y1 y2 y3 = w4 y4 = w3

. . .

yd w1 w2

. . .

wd Take largest ordered initial segment, v has maximum monochromatic degree (red) in the rest, exists not red uv, yiwi cannot be blue, all uwi are blue and wi is not in the ordered segment, u has higher mono degree than v.

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Thank you

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