V'Voloshin: Examples of Mixed Hypergraphs

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V.Voloshin Examples of Mixed Hypergraphs

• pictures speak louder than words...

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Drawing a graph G(V,E)

• V1,2,3,4,5
• E1,2,2,3,3,4,4,5,5,1,1,4,3,5

1
5
2
adjacent vertices
3
4
3
Coloring of G(V,E)

• V1,2,3,4,5
• E1,2,2,3,3,4,4,5,5,1,1,4,3,5

1
5
2
?(G)3
3
4
4
Drawing a hypergraph

• H(X,D), X1,2,3,4, D1,2,3,2,3,4,1,4
  D1,D2,D3,

2
D1
3
1
D3
D2
4
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Coloring a hypergraph

• H(X,D), X1,2,3,4, D1,2,3,2,3,4,1,4
  D1,D2,D3,

2
D1
3
1
D3
?(H)2
D2
4
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Drawing a mixed hypergraph H(X,C,D)

• X1,2,3,4, C1,2,4C1,
• D2,3,4,2,4D1,D2

3
C1
2
D2
D1
1
4
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Uncolorable mixed hypergraphs
D
the smallest uncolorable mixed hypergraph
1
2
C
violated C-edge
1
D1
C2
C1
uncolorable mixed hypergraph H(X,C,D)
2
violated D-edge
D2
D4
1
1
D3
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Coloring mixed hypergraph H(X,C,D) numbers
colors

• Proper 2-,3-,?-coloring
• Strict 2-coloring

1
C1
1
D1
D2
1
2
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Chromatic numbers
C
1
C
1
1
3
1
1
D
2
D
2
?(H)3
?(H)2
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Uniquely colorable mixed hypergraphs
H2(X,C,D)
H1(X,Ø,D)
C
1
D
1
D
2
D
D
2
D
D
C
D
C
3
2
1
1
R(H1)(0,0,1)
? ? 3
R(H2)(0,1,0,0,0)
? ? 2
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Chromatic polynomial
C
1
C
1
1
C
D
D
D
2
3
3
2
3
2
H
H
H
R (H)(0,3,0)
3 feasible partitions
3 2! strict 2-colorings
If we have ?2 colors, then the number P (H, ?)
of proper ?-colorings is 3 2! ?(?-1)/2! 3
?(?-1)3?(2)
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How to get a gap? Step 1
R (H )(0,r2,0,0)

• Construct a can

C
Feasible set S2
C
D
C
C


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How to get a gap? - Step 2
Construct
R (H ) (0, r2, r3, 0,0,0)


can
can
Feasible set S2,3


can


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How to get a gap? - Step 3

Each subset can



R (H)(0,r2, 0,r4,0,0,0,0)
Feasible set S2,4
We have broken it!!!




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The smallest mixed hypergraph with a gap, H2,4
(Jiang, Mubayi, Tuza, Voloshin, West)
D


Bi-edges
R(H2,4)(0,4,0,1,0,0)
D
C
S(H)2,4
C




C
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The smallest 3-uniform bihypergraph with a gap
(L.Gionfriddo, V.Voloshin, 2000)







R(H)(0,12,0,3,0,0,0), S(H)2,4
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Planar mixed hypergraph with a gap, H 2,4
(Kobler, Kundgen, 2001)
D

D
C
C
D
D
D




C
C
D

R(H2,4)(0,1,0,1,0,0), S(H)2,4
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THANK YOU!