fractional colorings and zykov products of graphs
TRANSCRIPT
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Fractional Coloringsand Zykov Products of graphs
Who? Nichole Schimanski
When? July 27, 2011
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GraphsA graph, G , consists of a vertex set, V (G ), and anedge set , E (G ).
V (G ) is any finite set
E (G ) is a set of unordered pairs of vertices
Example
Figure: Peterson graph
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GraphsA graph, G , consists of a vertex set, V (G ), and anedge set , E (G ).
V (G ) is any finite set
E (G ) is a set of unordered pairs of vertices
Example
Figure: Peterson graph
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Subgraphs
A subgraph H of a graph G is a graph such thatV (H) ⊆ V (G ) and E (H) ⊆ E (G ).
Example
Figure: Subgraph of the Peterson graph
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Subgraphs
A subgraph H of a graph G is a graph such thatV (H) ⊆ V (G ) and E (H) ⊆ E (G ).
Example
Figure: Subgraph of the Peterson graph
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Subgraphs
An induced subgraph, H, of G is a subgraph withproperty that any two vertices are adjacent in H if andonly if they are adjacent in G .
Example
Figure: Induced Subgraph of Peterson graph
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Subgraphs
An induced subgraph, H, of G is a subgraph withproperty that any two vertices are adjacent in H if andonly if they are adjacent in G .
Example
Figure: Induced Subgraph of Peterson graph
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Independent SetsA set of vertices, S, is said to be independent if thosevertices induce a graph with no edges.
Example
Figure: Independent set
The set of all independent sets of a graph G is denotedI (G ).
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Independent SetsA set of vertices, S, is said to be independent if thosevertices induce a graph with no edges.
Example
Figure: Independent set
The set of all independent sets of a graph G is denotedI (G ).
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Independent SetsA set of vertices, S, is said to be independent if thosevertices induce a graph with no edges.
Example
Figure: Independent set
The set of all independent sets of a graph G is denotedI (G ).
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Weighting I (S)A weighting of I (G ) is a function w : I (G )→ R≥0.
Example
a
e
d c
b
S w(S)
{a} 1/3{b} 1/3{c} 1/3{d} 1/3{e} 1/3{a,c} 1/3{a,d} 1/3{b,d} 1/3{b,e} 1/3{e,c} 1/3
Figure: C5 and a corresponding weighting
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Weighting I (S)A weighting of I (G ) is a function w : I (G )→ R≥0.
Example
a
e
d c
b
S w(S)
{a} 1/3{b} 1/3{c} 1/3{d} 1/3{e} 1/3{a,c} 1/3{a,d} 1/3{b,d} 1/3{b,e} 1/3{e,c} 1/3
Figure: C5 and a corresponding weighting
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Fractional k-coloring
A fractional k-coloring of a graph, G , is a weighting ofI (G ) such that∑
S∈I (G) w(S) = k ; and
For every v ∈ V (G ),∑S∈I (G)
v∈S
w(S) = w [v ] ≥ 1
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Fractional k-coloring
A fractional k-coloring of a graph, G , is a weighting ofI (G ) such that∑
S∈I (G) w(S) = k ; and
For every v ∈ V (G ),∑S∈I (G)
v∈S
w(S) = w [v ] ≥ 1
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Fractional k-coloring
Example
a
e
d c
b
S w(S)
{a} 1/3{b} 1/3{c} 1/3{d} 1/3{e} 1/3{a,c} 1/3{a,d} 1/3{b,d} 1/3{b,e} 1/3{e,c} 1/3
Figure: A fractional coloring of C5 with weight 10/3
∑S∈I (G) w(S) = 10/3
w [v ] = 1 for every v ∈ V (G )
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Fractional k-coloring
Example
a
e
d c
b
S w(S)
{a} 1/3{b} 1/3{c} 1/3{d} 1/3{e} 1/3{a,c} 1/3{a,d} 1/3{b,d} 1/3{b,e} 1/3{e,c} 1/3
Figure: A fractional coloring of C5 with weight 10/3∑S∈I (G) w(S) = 10/3
w [v ] = 1 for every v ∈ V (G )
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Fractional k-coloring
Example
a
e
d c
b
S w(S)
{a} 1/3{b} 1/3{c} 1/3{d} 1/3{e} 1/3{a,c} 1/3{a,d} 1/3{b,d} 1/3{b,e} 1/3{e,c} 1/3
Figure: A fractional coloring of C5 with weight 10/3∑S∈I (G) w(S) = 10/3
w [v ] = 1 for every v ∈ V (G )
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Fractional Chromatic Number
The fractional chromatic number, χf (G ), is theminimum possible weight of a fractional coloring.
Example
a
e
d c
b
S w(S)
{a} 0{b} 0{c} 0{d} 0{e} 0{a,c} 1/2{a,d} 1/2{b,d} 1/2{b,e} 1/2{e,c} 1/2
Figure: A weighting C5
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Fractional Chromatic Number
The fractional chromatic number, χf (G ), is theminimum possible weight of a fractional coloring.
Example
a
e
d c
b
S w(S)
{a} 0{b} 0{c} 0{d} 0{e} 0{a,c} 1/2{a,d} 1/2{b,d} 1/2{b,e} 1/2{e,c} 1/2
Figure: A weighting C5
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Fractional Chromatic Number
Example
a
e
d c
b
S w(S)
{a} 0{b} 0{c} 0{d} 0{e} 0{a,c} 1/2{a,d} 1/2{b,d} 1/2{b,e} 1/2{e,c} 1/2
Figure: A fractional 5/2-coloring of C5
∑S∈I (G) w(S) = 5/2
w [v ] = 1 for every v ∈ V (G )
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Fractional Chromatic Number
Example
a
e
d c
b
S w(S)
{a} 0{b} 0{c} 0{d} 0{e} 0{a,c} 1/2{a,d} 1/2{b,d} 1/2{b,e} 1/2{e,c} 1/2
Figure: A fractional 5/2-coloring of C5∑S∈I (G) w(S) = 5/2
w [v ] = 1 for every v ∈ V (G )
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Fractional Chromatic Number
Example
a
e
d c
b
S w(S)
{a} 0{b} 0{c} 0{d} 0{e} 0{a,c} 1/2{a,d} 1/2{b,d} 1/2{b,e} 1/2{e,c} 1/2
Figure: A fractional 5/2-coloring of C5∑S∈I (G) w(S) = 5/2
w [v ] = 1 for every v ∈ V (G )
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Fractional Chromatic Number
How do we know what the minimum is?
Linear Programming
Formulas
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Fractional Chromatic Number
How do we know what the minimum is?
Linear Programming
Formulas
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Fractional Chromatic Number
How do we know what the minimum is?
Linear Programming
Formulas
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Zykov Product of Graphs
The Zykov product Z(G1,G2, . . . ,Gn) of simple graphsG1,G2, . . . ,Gn is formed as follows.
Take the disjoint union of Gi
Example
Figure: Drawings of P2 and P3
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Zykov Product of Graphs
The Zykov product Z(G1,G2, . . . ,Gn) of simple graphsG1,G2, . . . ,Gn is formed as follows.
Take the disjoint union of Gi
Example
Figure: Drawings of P2 and P3
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Zykov Product of Graphs
The Zykov product Z(G1,G2, . . . ,Gn) of simple graphsG1,G2, . . . ,Gn is formed as follows.
Take the disjoint union of Gi
Example
Figure: Drawings of P2 and P3
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Zykov Product of Graphs
The Zykov product Z(G1,G2, . . . ,Gn) of simple graphsG1,G2, . . . ,Gn is formed as follows.
Take the disjoint union of Gi
Example
Figure: Drawings of P2 and P3
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Zykov Product of Graphs
The Zykov product Z(G1,G2, . . . ,Gn) of simple graphsG1,G2, . . . ,Gn is formed as follows.
Take the disjoint union of Gi
For each (x1, . . . , xn) ∈ V (G1)× V (G2)× . . .× V (Gn)add a new vertex adjacent to the vertices {x1, . . . , xn}
Example
Figure: Constructing Z(P2,P3)
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Zykov Product of Graphs
The Zykov product Z(G1,G2, . . . ,Gn) of simple graphsG1,G2, . . . ,Gn is formed as follows.
Take the disjoint union of Gi
For each (x1, . . . , xn) ∈ V (G1)× V (G2)× . . .× V (Gn)add a new vertex adjacent to the vertices {x1, . . . , xn}
Example
Figure: Constructing Z(P2,P3)
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Zykov Product of Graphs
The Zykov product Z(G1,G2, . . . ,Gn) of simple graphsG1,G2, . . . ,Gn is formed as follows.
Take the disjoint union of Gi
For each (x1, . . . , xn) ∈ V (G1)× V (G2)× . . .× V (Gn)add a new vertex adjacent to the vertices {x1, . . . , xn}
Example
Figure: Z(P2,P3)
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Zykov Graphs
The Zykov graphs, Zn, are formed as follows:
Set Z1 as a single vertex
Define Zn := Z(Z1, ...,Zn−1) for all n ≥ 2
Figure: Drawing of Z1
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Zykov Graphs
The Zykov graphs, Zn, are formed as follows:
Set Z1 as a single vertex
Define Zn := Z(Z1, ...,Zn−1) for all n ≥ 2
Figure: Drawing of Z1
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Zykov Graphs
The Zykov graphs, Zn, are formed as follows:
Set Z1 as a single vertex
Define Zn := Z(Z1, ...,Zn−1) for all n ≥ 2
Figure: Drawing of Z1
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Zykov Graphs
The Zykov graphs, Zn, are formed as follows:
Set Z1 as a single vertex
Define Zn := Z(Z1, ...,Zn−1) for all n ≥ 2
Figure: Drawings of Z1 and Z2
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Zykov Graphs
The Zykov graphs, Zn, are formed as follows:
Set Z1 as a single vertex
Define Zn := Z(Z1, ...,Zn−1) for all n ≥ 2
Figure: Drawings of Z1, Z2, and Z3
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Zykov Graphs
The Zykov graphs, Zn, are formed as follows:
Set Z1 as a single vertex
Define Zn := Z(Z1, ...,Zn−1) for all n ≥ 2
Figure: Drawing of Z4
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Jacobs’ Conjecture
Corollary For n ≥ 1,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Example
χf (Z1) = 1
χf (Z2) = 1 + 11 = 2
χf (Z3) = 2 + 12 = 5
2
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Jacobs’ Conjecture
Corollary For n ≥ 1,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Example
χf (Z1) = 1
χf (Z2) = 1 + 11 = 2
χf (Z3) = 2 + 12 = 5
2
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Jacobs’ Conjecture
Corollary For n ≥ 1,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Example
χf (Z1) = 1
χf (Z2) = 1 + 11 = 2
χf (Z3) = 2 + 12 = 5
2
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Jacobs’ Conjecture
Corollary For n ≥ 1,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Example
χf (Z1) = 1
χf (Z2) = 1 + 11 = 2
χf (Z3) = 2 + 12 = 5
2
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Jacobs’ Conjecture
Corollary For n ≥ 1,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Example
χf (Z1) = 1
χf (Z2) = 1 + 11 = 2
χf (Z3) = 2 + 12 = 5
2
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Verifying χf (C5)
Notice that
→
Figure: Z3 and C5
So, χf (Z3) = χf (C5) = 5/2
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Verifying χf (C5)
Notice that
→
Figure: Z3 and C5
So, χf (Z3) = χf (C5) = 5/2
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Verifying χf (C5)
Notice that
→
Figure: Z3 and C5
So, χf (Z3) = χf (C5) = 5/2
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The Main Result: Theorem 1
Theorem For n ≥ 2, let G1, . . . ,Gn be graphs. SetG := Z(G1, . . . ,Gn) and χi = χf (Gi ). Suppose alsothat the graphs Gi are numbered such that χi ≤ χi+1.Then
χf (G ) = max
(χn, 2 +
n∑i=2
n∏k=i
(1− 1
χk
))
Example
χf (Z(P2,P3)) = max
(2, 2 +
(1− 1
2
))= max(2,
5
2)
=5
2.
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The Main Result: Theorem 1Theorem For n ≥ 2, let G1, . . . ,Gn be graphs. Set
G := Z(G1, . . . ,Gn) and χi = χf (Gi ). Suppose alsothat the graphs Gi are numbered such that χi ≤ χi+1.Then
χf (G ) = max
(χn, 2 +
n∑i=2
n∏k=i
(1− 1
χk
))
Example
χf (Z(P2,P3)) = max
(2, 2 +
(1− 1
2
))= max(2,
5
2)
=5
2.
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The Main Result: Theorem 1Theorem For n ≥ 2, let G1, . . . ,Gn be graphs. Set
G := Z(G1, . . . ,Gn) and χi = χf (Gi ). Suppose alsothat the graphs Gi are numbered such that χi ≤ χi+1.Then
χf (G ) = max
(χn, 2 +
n∑i=2
n∏k=i
(1− 1
χk
))
Example
χf (Z(P2,P3)) = max
(2, 2 +
(1− 1
2
))= max(2,
5
2)
=5
2.
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Lower Bound: χf (G ) ≥ max (χn, f (n))
Lemma The fractional chromatic number of a subgraph, H, is atmost equal to the fractional chromatic number of agraph, G .
Conclusion χf (G ) ≥ χn
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Lower Bound: χf (G ) ≥ max (χn, f (n))
Lemma The fractional chromatic number of a subgraph, H, is atmost equal to the fractional chromatic number of agraph, G .
Conclusion χf (G ) ≥ χn
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Lower Bound: χf (G ) ≥ max (χn, f (n))
Lemma The fractional chromatic number of a subgraph, H, is atmost equal to the fractional chromatic number of agraph, G .
Conclusion χf (G ) ≥ χn
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Lower Bound: χf (G ) ≥ max (χn, f (n))
Lemma Let G be a graph and w a weighting of X ⊆ I (G ).Then, for every induced subgraph H of G , there existsx ∈ V (H) such that
w [x ] ≤ 1
χf (H)
∑S∈X
w(S).
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Lower Bound: χf (G ) ≥ max (χn, f (n))
Start with w , a χf -coloring of G and x1 ∈ V (G1).
Construct F1 = {S ∈ I (G ) : x1 ∈ S} with the property∑S∈F1
w(S) = w [x1] ≥ 1.
Construct F2 = {S ∈ I (G ) : S ∩ {x1, x2} 6= ∅} with
the property∑
S∈F2w(S) ≥ 1 +
(1− 1
χ2
)∑S∈F1
w(S).
![Page 55: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/55.jpg)
Lower Bound: χf (G ) ≥ max (χn, f (n))
Start with w , a χf -coloring of G and x1 ∈ V (G1).
Construct F1 = {S ∈ I (G ) : x1 ∈ S} with the property∑S∈F1
w(S) = w [x1] ≥ 1.
Construct F2 = {S ∈ I (G ) : S ∩ {x1, x2} 6= ∅} with
the property∑
S∈F2w(S) ≥ 1 +
(1− 1
χ2
)∑S∈F1
w(S).
![Page 56: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/56.jpg)
Lower Bound: χf (G ) ≥ max (χn, f (n))
Start with w , a χf -coloring of G and x1 ∈ V (G1).
Construct F1 = {S ∈ I (G ) : x1 ∈ S} with the property∑S∈F1
w(S) = w [x1] ≥ 1.
Construct F2 = {S ∈ I (G ) : S ∩ {x1, x2} 6= ∅} with
the property∑
S∈F2w(S) ≥ 1 +
(1− 1
χ2
)∑S∈F1
w(S).
![Page 57: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/57.jpg)
Lower Bound: χf (G ) ≥ max (χn, f (n))
Continue this process so that for all k ∈ {1, ..., n},
Fk = {S ∈ I (G ) : S ∩ {x1, . . . , xk} 6= ∅} with the
property∑
S∈Fkw(S) ≥ 1 +
(1− 1
χk
)∑S∈Fk−1
w(S)
It follows that∑S∈Fn
w(S) ≥ 1 +n∑
i=2
n∏k=i
(1− 1
χk
)= f (n)− 1.
![Page 58: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/58.jpg)
Lower Bound: χf (G ) ≥ max (χn, f (n))
Continue this process so that for all k ∈ {1, ..., n},
Fk = {S ∈ I (G ) : S ∩ {x1, . . . , xk} 6= ∅} with the
property∑
S∈Fkw(S) ≥ 1 +
(1− 1
χk
)∑S∈Fk−1
w(S)
It follows that∑S∈Fn
w(S) ≥ 1 +n∑
i=2
n∏k=i
(1− 1
χk
)= f (n)− 1.
![Page 59: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/59.jpg)
Lower Bound: χf (G ) ≥ max (χn, f (n))
Continue this process so that for all k ∈ {1, ..., n},
Fk = {S ∈ I (G ) : S ∩ {x1, . . . , xk} 6= ∅} with the
property∑
S∈Fkw(S) ≥ 1 +
(1− 1
χk
)∑S∈Fk−1
w(S)
It follows that∑S∈Fn
w(S) ≥ 1 +n∑
i=2
n∏k=i
(1− 1
χk
)= f (n)− 1.
![Page 60: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/60.jpg)
Lower Bound: χf (G ) ≥ max (χn, f (n))
Conclusion χf (G ) ≥ f (n)
![Page 61: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/61.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))Special Sets and Cool Weightings
Special Sets Let M (G ) ⊂ I (G ) be the set of all maximalindependent sets of G and for each i ∈ {1, ..., n},
Fi := {S ∈M (G )|S ∩ V (Gj) = ∅ if and only if j < i}
Weightings
wi : I (Gi )→ R≥0, a χf (Gi )-coloring of each Gi
pi : I (Gi )→ R≥0 where pi := wi (S)/χi
p : ∪ni=1Fi → R≥0 where p(S) :=∏n
i=1 pi (S ∩ V (Gi ))
![Page 62: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/62.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))Special Sets and Cool Weightings
Special Sets Let M (G ) ⊂ I (G ) be the set of all maximalindependent sets of G
and for each i ∈ {1, ..., n},
Fi := {S ∈M (G )|S ∩ V (Gj) = ∅ if and only if j < i}
Weightings
wi : I (Gi )→ R≥0, a χf (Gi )-coloring of each Gi
pi : I (Gi )→ R≥0 where pi := wi (S)/χi
p : ∪ni=1Fi → R≥0 where p(S) :=∏n
i=1 pi (S ∩ V (Gi ))
![Page 63: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/63.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))Special Sets and Cool Weightings
Special Sets Let M (G ) ⊂ I (G ) be the set of all maximalindependent sets of G and for each i ∈ {1, ..., n},
Fi := {S ∈M (G )|S ∩ V (Gj) = ∅ if and only if j < i}
Weightings
wi : I (Gi )→ R≥0, a χf (Gi )-coloring of each Gi
pi : I (Gi )→ R≥0 where pi := wi (S)/χi
p : ∪ni=1Fi → R≥0 where p(S) :=∏n
i=1 pi (S ∩ V (Gi ))
![Page 64: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/64.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))Special Sets and Cool Weightings
Special Sets Let M (G ) ⊂ I (G ) be the set of all maximalindependent sets of G and for each i ∈ {1, ..., n},
Fi := {S ∈M (G )|S ∩ V (Gj) = ∅ if and only if j < i}
Weightings
wi : I (Gi )→ R≥0, a χf (Gi )-coloring of each Gi
pi : I (Gi )→ R≥0 where pi := wi (S)/χi
p : ∪ni=1Fi → R≥0 where p(S) :=∏n
i=1 pi (S ∩ V (Gi ))
![Page 65: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/65.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))Special Sets and Cool Weightings
Special Sets Let M (G ) ⊂ I (G ) be the set of all maximalindependent sets of G and for each i ∈ {1, ..., n},
Fi := {S ∈M (G )|S ∩ V (Gj) = ∅ if and only if j < i}
Weightings
wi : I (Gi )→ R≥0, a χf (Gi )-coloring of each Gi
pi : I (Gi )→ R≥0 where pi := wi (S)/χi
p : ∪ni=1Fi → R≥0 where p(S) :=∏n
i=1 pi (S ∩ V (Gi ))
![Page 66: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/66.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))Special Sets and Cool Weightings
Special Sets Let M (G ) ⊂ I (G ) be the set of all maximalindependent sets of G and for each i ∈ {1, ..., n},
Fi := {S ∈M (G )|S ∩ V (Gj) = ∅ if and only if j < i}
Weightings
wi : I (Gi )→ R≥0, a χf (Gi )-coloring of each Gi
pi : I (Gi )→ R≥0 where pi := wi (S)/χi
p : ∪ni=1Fi → R≥0 where p(S) :=∏n
i=1 pi (S ∩ V (Gi ))
![Page 67: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/67.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))The Final Weighting
FinalWeighting
We construct a fractional max(χn, f (n))-coloring of Gdefined by the weighting
w(S) =
(χi − χi−1)p(S), S ∈ Fi
max(0, f (n)− χn), S = V0
0, otherwise
![Page 68: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/68.jpg)
Upper Bound: χf (G ) ≤ max (χn, f (n))The Final Weighting works!
We can show,∑S∈I (G) w(S) = max(χn, f (n))
w [x ] ≥ 1 for all x ∈ V (G )
So, w is a fractional max(χn, f (n))-coloring of G .
Conclusion χf (G ) ≤ max (χn, f (n))
![Page 69: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/69.jpg)
Results
Theorem For n ≥ 2, let G1, . . . ,Gn be graph. Suppose also thatthe graphs Gi are numbered such that χi ≤ χi+1. Then
χf (Z(G1, . . . ,Gn)) = max
(χn, 2 +
n∑i=2
n∏k=i
(1− 1
χk
))
Corollary For every n ≥ 2,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
![Page 70: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/70.jpg)
Jacobs’ Conjecture - Proved!
Corollary For every n ≥ 2,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Proof. By induction on n ≥ 2, we proveχn+1 = f (n) = χn + χ−1n .
Base Case: χf (Z1) = 1 and f (1) = 2 = χ2
Inductive Hypothesis: Suppose χn = f (n − 1).
![Page 71: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/71.jpg)
Jacobs’ Conjecture - Proved!
Corollary For every n ≥ 2,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Proof. By induction on n ≥ 2, we proveχn+1 = f (n) = χn + χ−1n .
Base Case: χf (Z1) = 1 and f (1) = 2 = χ2
Inductive Hypothesis: Suppose χn = f (n − 1).
![Page 72: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/72.jpg)
Jacobs’ Conjecture - Proved!
Corollary For every n ≥ 2,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Proof. By induction on n ≥ 2, we proveχn+1 = f (n) = χn + χ−1n .
Base Case: χf (Z1) = 1 and f (1) = 2 = χ2
Inductive Hypothesis: Suppose χn = f (n − 1).
![Page 73: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/73.jpg)
Jacobs’ Conjecture - Proved!
Corollary For every n ≥ 2,
χf (Zn+1) = χf (Zn) +1
χf (Zn)
Proof. By induction on n ≥ 2, we proveχn+1 = f (n) = χn + χ−1n .
Base Case: χf (Z1) = 1 and f (1) = 2 = χ2
Inductive Hypothesis: Suppose χn = f (n − 1).
![Page 74: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/74.jpg)
Jacobs’ Conjecture - Proved!
Proof.
Inductive Hypothesis: Suppose χn = f (n − 1).
Then
f (n) = 2 +n∑
i=2
∏k≥i
(1− 1
χk
)
= 2 +
(1− 1
χn
)· (f (n − 1)− 1)
= χn +1
χn
Since χn+1 = max(χn, f (n)), we haveχn+1 = χn + 1
χn.
![Page 75: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/75.jpg)
Jacobs’ Conjecture - Proved!
Proof.
Inductive Hypothesis: Suppose χn = f (n − 1). Then
f (n) = 2 +n∑
i=2
∏k≥i
(1− 1
χk
)
= 2 +
(1− 1
χn
)· (f (n − 1)− 1)
= χn +1
χn
Since χn+1 = max(χn, f (n)), we haveχn+1 = χn + 1
χn.
![Page 76: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/76.jpg)
Jacobs’ Conjecture - Proved!
Proof.
Inductive Hypothesis: Suppose χn = f (n − 1). Then
f (n) = 2 +n∑
i=2
∏k≥i
(1− 1
χk
)
= 2 +
(1− 1
χn
)· (f (n − 1)− 1)
= χn +1
χn
Since χn+1 = max(χn, f (n)), we haveχn+1 = χn + 1
χn.
![Page 77: Fractional Colorings and Zykov Products of graphs](https://reader030.vdocuments.mx/reader030/viewer/2022012700/61a340d646760c427876209d/html5/thumbnails/77.jpg)
Questions?