Coloring Clean and K4-free Circle Graphskgmilans/research/circ-graph/... · Coloring Clean and K...
Transcript of Coloring Clean and K4-free Circle Graphskgmilans/research/circ-graph/... · Coloring Clean and K...
Coloring Clean and K4-free Circle Graphs
Kevin G. Milans ([email protected])Joint with A.V. Kostochka
University of South Carolina
Atlanta Lecture Series in Discrete Mathematics: IIIAtlanta, GA
2011 April 16
Circle Graphs
DefinitionA circle graph is the intersection graph of chords in a circle.
Example
I They arise in sorting problems,
I questions in topological graph theory,
I and VLSI design.
I We may assume the endpoints are distinct.
Circle Graphs
DefinitionA circle graph is the intersection graph of chords in a circle.
Example
I They arise in sorting problems,
I questions in topological graph theory,
I and VLSI design.
I We may assume the endpoints are distinct.
Circle Graphs
DefinitionA circle graph is the intersection graph of chords in a circle.
Example
I They arise in sorting problems,
I questions in topological graph theory,
I and VLSI design.
I We may assume the endpoints are distinct.
Circle Graphs
DefinitionA circle graph is the intersection graph of chords in a circle.
Example
I They arise in sorting problems,
I questions in topological graph theory,
I and VLSI design.
I We may assume the endpoints are distinct.
Circle Graphs
DefinitionA circle graph is the intersection graph of chords in a circle.
Example
I They arise in sorting problems,
I questions in topological graph theory,
I and VLSI design.
I We may assume the endpoints are distinct.
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:
I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:
I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )
I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:
I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )
I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:
I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:
I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:
I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:I Keil (1993): domination number
I Garey–Johnson–Miller–Papadmitriou (1980):chromatic number χ(G ).
I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).
I Keil–Stewart (2006): clique covering number
Complexity of Circle Graphs
I Spinrad (1994): there is an O(n2)-time algorithm that decidesif G is a circle graph.
I Some hard problems become easy for circle graphs:
I Gavril (1973): clique number ω(G )I Gavril (1973): independence number α(G )I Kloks (1996): O(n3)-time algorithm for treewidth
I Other problems remain NP-hard:I Keil (1993): domination numberI Garey–Johnson–Miller–Papadmitriou (1980):
chromatic number χ(G ).I Keil–Stewart (2006): clique covering number
Chromatic Numbers of Circle Graphs
I Important for applications to sorting and VLSI.
TheoremIf G is a circle graph with ω(G ) = k, then
I χ(G ) ≤ k2 · 4k (Gyarfas (1985)).
I χ(G ) ≤ k3 · 2k (Kostochka (1988)).
I χ(G ) ≤ 50 · 2k (Kostochka–Kratochvıl (1997)).
Theorem (Kostochka (1988))
For infinitely many k, there is a circle graph G with ω(G ) = k and
χ(G ) ≥(
1
2− 1
ln k
)k ln k.
I This exponential gap has persisted for 25 years.
Chromatic Numbers of Circle Graphs
I Important for applications to sorting and VLSI.
TheoremIf G is a circle graph with ω(G ) = k, then
I χ(G ) ≤ k2 · 4k (Gyarfas (1985)).
I χ(G ) ≤ k3 · 2k (Kostochka (1988)).
I χ(G ) ≤ 50 · 2k (Kostochka–Kratochvıl (1997)).
Theorem (Kostochka (1988))
For infinitely many k, there is a circle graph G with ω(G ) = k and
χ(G ) ≥(
1
2− 1
ln k
)k ln k.
I This exponential gap has persisted for 25 years.
Chromatic Numbers of Circle Graphs
I Important for applications to sorting and VLSI.
TheoremIf G is a circle graph with ω(G ) = k, then
I χ(G ) ≤ k2 · 4k (Gyarfas (1985)).
I χ(G ) ≤ k3 · 2k (Kostochka (1988)).
I χ(G ) ≤ 50 · 2k (Kostochka–Kratochvıl (1997)).
Theorem (Kostochka (1988))
For infinitely many k, there is a circle graph G with ω(G ) = k and
χ(G ) ≥(
1
2− 1
ln k
)k ln k.
I This exponential gap has persisted for 25 years.
Chromatic Numbers of Circle Graphs
I Important for applications to sorting and VLSI.
TheoremIf G is a circle graph with ω(G ) = k, then
I χ(G ) ≤ k2 · 4k (Gyarfas (1985)).
I χ(G ) ≤ k3 · 2k (Kostochka (1988)).
I χ(G ) ≤ 50 · 2k (Kostochka–Kratochvıl (1997)).
Theorem (Kostochka (1988))
For infinitely many k, there is a circle graph G with ω(G ) = k and
χ(G ) ≥(
1
2− 1
ln k
)k ln k.
I This exponential gap has persisted for 25 years.
Chromatic Numbers of Circle Graphs
I Important for applications to sorting and VLSI.
TheoremIf G is a circle graph with ω(G ) = k, then
I χ(G ) ≤ k2 · 4k (Gyarfas (1985)).
I χ(G ) ≤ k3 · 2k (Kostochka (1988)).
I χ(G ) ≤ 50 · 2k (Kostochka–Kratochvıl (1997)).
Theorem (Kostochka (1988))
For infinitely many k, there is a circle graph G with ω(G ) = k and
χ(G ) ≥(
1
2− 1
ln k
)k ln k.
I This exponential gap has persisted for 25 years.
Chromatic Numbers of Circle Graphs
I Important for applications to sorting and VLSI.
TheoremIf G is a circle graph with ω(G ) = k, then
I χ(G ) ≤ k2 · 4k (Gyarfas (1985)).
I χ(G ) ≤ k3 · 2k (Kostochka (1988)).
I χ(G ) ≤ 50 · 2k (Kostochka–Kratochvıl (1997)).
Theorem (Kostochka (1988))
For infinitely many k, there is a circle graph G with ω(G ) = k and
χ(G ) ≥(
1
2− 1
ln k
)k ln k.
I This exponential gap has persisted for 25 years.
Chromatic Numbers of Circle Graphs
I Important for applications to sorting and VLSI.
TheoremIf G is a circle graph with ω(G ) = k, then
I χ(G ) ≤ k2 · 4k (Gyarfas (1985)).
I χ(G ) ≤ k3 · 2k (Kostochka (1988)).
I χ(G ) ≤ 50 · 2k (Kostochka–Kratochvıl (1997)).
Theorem (Kostochka (1988))
For infinitely many k, there is a circle graph G with ω(G ) = k and
χ(G ) ≥(
1
2− 1
ln k
)k ln k.
I This exponential gap has persisted for 25 years.
Chromatic Numbers of K3-free Circle Graphs
TheoremIf G is a triangle-free circle graph, then
I χ(G ) ≤ 8 (Karapetyan (1985)).
I χ(G ) ≤ 5 (Kostochka (1988)).
Theorem (Ageev (1996))
There is a triangle-free circle graph with chromatic number 5.
I Sharp bounds are known only for triangle-free circle graphs.
Chromatic Numbers of K3-free Circle Graphs
TheoremIf G is a triangle-free circle graph, then
I χ(G ) ≤ 8 (Karapetyan (1985)).
I χ(G ) ≤ 5 (Kostochka (1988)).
Theorem (Ageev (1996))
There is a triangle-free circle graph with chromatic number 5.
I Sharp bounds are known only for triangle-free circle graphs.
Chromatic Numbers of K3-free Circle Graphs
TheoremIf G is a triangle-free circle graph, then
I χ(G ) ≤ 8 (Karapetyan (1985)).
I χ(G ) ≤ 5 (Kostochka (1988)).
Theorem (Ageev (1996))
There is a triangle-free circle graph with chromatic number 5.
I Sharp bounds are known only for triangle-free circle graphs.
Chromatic Numbers of K3-free Circle Graphs
TheoremIf G is a triangle-free circle graph, then
I χ(G ) ≤ 8 (Karapetyan (1985)).
I χ(G ) ≤ 5 (Kostochka (1988)).
Theorem (Ageev (1996))
There is a triangle-free circle graph with chromatic number 5.
I Sharp bounds are known only for triangle-free circle graphs.
Chromatic Numbers of K3-free Circle Graphs
TheoremIf G is a triangle-free circle graph, then
I χ(G ) ≤ 8 (Karapetyan (1985)).
I χ(G ) ≤ 5 (Kostochka (1988)).
Theorem (Ageev (1996))
There is a triangle-free circle graph with chromatic number 5.
I Sharp bounds are known only for triangle-free circle graphs.
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Overlap Graphs
Definition
I Two sets overlap if they have non-empty intersection andneither is contained in the other.
I An overlap graph is a the overlap graph of a set of closedintervals in R.
FactG is a circle graph if and only if G is an overlap graph.
Example
Clean Circle Graphs
Definition
I A set of intervals is clean if it does not contain the following:
I A circle graph is clean if it is the overlap graph of a clean setof intervals.
TheoremIf G is a clean circle graph with ω(G ) = k, then χ(G ) ≤ 2k − 1.
Clean Circle Graphs
Definition
I A set of intervals is clean if it does not contain the following:
I A circle graph is clean if it is the overlap graph of a clean setof intervals.
TheoremIf G is a clean circle graph with ω(G ) = k, then χ(G ) ≤ 2k − 1.
Clean Circle Graphs
Definition
I A set of intervals is clean if it does not contain the following:
I A circle graph is clean if it is the overlap graph of a clean setof intervals.
TheoremIf G is a clean circle graph with ω(G ) = k, then χ(G ) ≤ 2k − 1.
A Coloring Strategy
c
I Given a family intervals X , how do we color the overlap graph?
I One natural strategy: pick a point c ∈ R.
I This partitions X into X<c , X c , and X>c .
I Color X<c ∪ X c and X c ∪ X>c inductively.
I Hope the colorings agree on X c?
I We need to control the coloring on X c .
A Coloring Strategy
c
I Given a family intervals X , how do we color the overlap graph?
I One natural strategy: pick a point c ∈ R.
I This partitions X into X<c , X c , and X>c .
I Color X<c ∪ X c and X c ∪ X>c inductively.
I Hope the colorings agree on X c?
I We need to control the coloring on X c .
A Coloring Strategy
c
I Given a family intervals X , how do we color the overlap graph?
I One natural strategy: pick a point c ∈ R.
I This partitions X into X<c , X c , and X>c .
I Color X<c ∪ X c and X c ∪ X>c inductively.
I Hope the colorings agree on X c?
I We need to control the coloring on X c .
A Coloring Strategy
c
I Given a family intervals X , how do we color the overlap graph?
I One natural strategy: pick a point c ∈ R.
I This partitions X into X<c , X c , and X>c .
I Color X<c ∪ X c and X c ∪ X>c inductively.
I Hope the colorings agree on X c?
I We need to control the coloring on X c .
A Coloring Strategy
c
I Given a family intervals X , how do we color the overlap graph?
I One natural strategy: pick a point c ∈ R.
I This partitions X into X<c , X c , and X>c .
I Color X<c ∪ X c and X c ∪ X>c inductively.
I Hope the colorings agree on X c?
I We need to control the coloring on X c .
A Coloring Strategy
c
I Given a family intervals X , how do we color the overlap graph?
I One natural strategy: pick a point c ∈ R.
I This partitions X into X<c , X c , and X>c .
I Color X<c ∪ X c and X c ∪ X>c inductively.
I Hope the colorings agree on X c?
I We need to control the coloring on X c .
A Coloring Strategy
c
I Given a family intervals X , how do we color the overlap graph?
I One natural strategy: pick a point c ∈ R.
I This partitions X into X<c , X c , and X>c .
I Color X<c ∪ X c and X c ∪ X>c inductively.
I Hope the colorings agree on X c?
I We need to control the coloring on X c .
The Canonical Coloring
w
x
w ≤ x
w x
w ≤ x
w
x
w ||x
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
w
x
w ≤ x
w x
w ≤ x
w
x
w ||x
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
w
x
w ≤ x
w x
w ≤ x
w
x
w ||x
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
12
34
4
4
5
56
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
1
2
34
4
4
5
56
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
12
34
4
4
5
56
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
12
3
4
4
4
5
56
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
12
34
4
4
5
56
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
12
34
4
4
5
5
6
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
12
34
4
4
5
56
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
1
12
34
4
4
5
56
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
13
31
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
1
3
3
1
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
1
3
3
12
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
13
31
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
13
31
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
13
31
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
13
31
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
13
31
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
The Canonical Coloring
1
22
13
31
2
4
I The endpoint order on a family of intervals puts [a, b] ≤ [c , d ]if a ≤ c and b ≤ d .
I The canonical coloring on a family of intervals colors eachinterval with its height in the endpoint order.
I Each color class is a chain by inclusion.
Proposition
When all intervals share a common point, the overlap graph isperfect and the canonical coloring is optimal.
Good Colorings
x
x
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.
I Canonical on R[x ]: f (u) = f (v)I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings
x
x
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.
I Canonical on R[x ]: f (u) = f (v)I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings
x
x
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.
I Canonical on R[x ]: f (u) = f (v)I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings
x
y
vu
x
y
vu
x
y
vu
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.
I Canonical on R[x ]: f (u) = f (v)I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings
x
y
vu
x
y
vu
x
y
vu
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.I Canonical on R[x ]: f (u) = f (v)
I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings
x
y
vu
x
y
vu
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.I Canonical on R[x ]: f (u) = f (v)I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings
x
y
vu
x
y
vu
x
y
vu
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.I Canonical on R[x ]: f (u) = f (v)I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings
x
y
vu
x
y
vu
I The closed right-neighborhood of x , denoted R[x ], consists ofx and all neighbors of x that overlap x to the right.
I A coloring is good if it is canonical on every closedright-neighborhood.
I Not all families of intervals have good colorings.I Canonical on R[x ]: f (u) = f (v)I Canonical on R[y ]: f (u) 6= f (v)
Good Colorings of Clean Families
TheoremIf X is a clean family of intervals whose overlap graph has cliquenumber k, then X has a good coloring using at most 2k − 1 colors.
I Most cases need only k + 1 colors.
TheoremThere is a clean family of intervals whose overlap graph has cliquenumber k for which every good coloring uses at least 2k − 1 colors.
I Good colorings are stronger than proper colorings.
I If we require only that the coloring is proper, how many colorsare needed?
I k = 2: 3 colors are sufficient and sometimes necessary.
I k = 3: 5 colors are sufficient and 4 are sometimes necessary.
Good Colorings of Clean Families
TheoremIf X is a clean family of intervals whose overlap graph has cliquenumber k, then X has a good coloring using at most 2k − 1 colors.
I Most cases need only k + 1 colors.
TheoremThere is a clean family of intervals whose overlap graph has cliquenumber k for which every good coloring uses at least 2k − 1 colors.
I Good colorings are stronger than proper colorings.
I If we require only that the coloring is proper, how many colorsare needed?
I k = 2: 3 colors are sufficient and sometimes necessary.
I k = 3: 5 colors are sufficient and 4 are sometimes necessary.
Good Colorings of Clean Families
TheoremIf X is a clean family of intervals whose overlap graph has cliquenumber k, then X has a good coloring using at most 2k − 1 colors.
I Most cases need only k + 1 colors.
TheoremThere is a clean family of intervals whose overlap graph has cliquenumber k for which every good coloring uses at least 2k − 1 colors.
I Good colorings are stronger than proper colorings.
I If we require only that the coloring is proper, how many colorsare needed?
I k = 2: 3 colors are sufficient and sometimes necessary.
I k = 3: 5 colors are sufficient and 4 are sometimes necessary.
Good Colorings of Clean Families
TheoremIf X is a clean family of intervals whose overlap graph has cliquenumber k, then X has a good coloring using at most 2k − 1 colors.
I Most cases need only k + 1 colors.
TheoremThere is a clean family of intervals whose overlap graph has cliquenumber k for which every good coloring uses at least 2k − 1 colors.
I Good colorings are stronger than proper colorings.
I If we require only that the coloring is proper, how many colorsare needed?
I k = 2: 3 colors are sufficient and sometimes necessary.
I k = 3: 5 colors are sufficient and 4 are sometimes necessary.
Good Colorings of Clean Families
TheoremIf X is a clean family of intervals whose overlap graph has cliquenumber k, then X has a good coloring using at most 2k − 1 colors.
I Most cases need only k + 1 colors.
TheoremThere is a clean family of intervals whose overlap graph has cliquenumber k for which every good coloring uses at least 2k − 1 colors.
I Good colorings are stronger than proper colorings.
I If we require only that the coloring is proper, how many colorsare needed?
I k = 2: 3 colors are sufficient and sometimes necessary.
I k = 3: 5 colors are sufficient and 4 are sometimes necessary.
Good Colorings of Clean Families
TheoremIf X is a clean family of intervals whose overlap graph has cliquenumber k, then X has a good coloring using at most 2k − 1 colors.
I Most cases need only k + 1 colors.
TheoremThere is a clean family of intervals whose overlap graph has cliquenumber k for which every good coloring uses at least 2k − 1 colors.
I Good colorings are stronger than proper colorings.
I If we require only that the coloring is proper, how many colorsare needed?
I k = 2: 3 colors are sufficient and sometimes necessary.
I k = 3: 5 colors are sufficient and 4 are sometimes necessary.
Good Colorings of Clean Families
TheoremIf X is a clean family of intervals whose overlap graph has cliquenumber k, then X has a good coloring using at most 2k − 1 colors.
I Most cases need only k + 1 colors.
TheoremThere is a clean family of intervals whose overlap graph has cliquenumber k for which every good coloring uses at least 2k − 1 colors.
I Good colorings are stronger than proper colorings.
I If we require only that the coloring is proper, how many colorsare needed?
I k = 2: 3 colors are sufficient and sometimes necessary.
I k = 3: 5 colors are sufficient and 4 are sometimes necessary.
Coloring Clean Families
x
z
y
y
v
v
x
z
yvv
y
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I Find appropriate intervals x , y , z , and v . Delete y .
I With y removed, z and v are inclusion-maximal.
I Extend v .
I Obtain a good coloring inductively.
I Restore v .
I Restore y .
Coloring Clean Families
x
z
y
yv
v
x
z
yvv
y
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I Find appropriate intervals x , y , z , and v . Delete y .
I With y removed, z and v are inclusion-maximal.
I Extend v .
I Obtain a good coloring inductively.
I Restore v .
I Restore y .
Coloring Clean Families
x
z
y
yv
v
x
z
yvv
y
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I Find appropriate intervals x , y , z , and v . Delete y .
I With y removed, z and v are inclusion-maximal.
I Extend v .
I Obtain a good coloring inductively.
I Restore v .
I Restore y .
Coloring Clean Families
x
z
y
y
v
v
x
z
yvv
y
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I Find appropriate intervals x , y , z , and v . Delete y .
I With y removed, z and v are inclusion-maximal.
I Extend v .
I Obtain a good coloring inductively.
I Restore v .
I Restore y .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I Find appropriate intervals x , y , z , and v . Delete y .
I With y removed, z and v are inclusion-maximal.
I Extend v .
I Obtain a good coloring inductively.
I Restore v .
I Restore y .
Coloring Clean Families
x
z
yyvv
x
z
y
v
v
y
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I Find appropriate intervals x , y , z , and v . Delete y .
I With y removed, z and v are inclusion-maximal.
I Extend v .
I Obtain a good coloring inductively.
I Restore v .
I Restore y .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I Find appropriate intervals x , y , z , and v . Delete y .
I With y removed, z and v are inclusion-maximal.
I Extend v .
I Obtain a good coloring inductively.
I Restore v .
I Restore y .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I How should we color y?
I R[x ] canonical: y must get same color as z (color 2 ).
I Let c be a point just to the right of z .
I Let d be a point just to the right of y .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I How should we color y?
I R[x ] canonical: y must get same color as z (color 2 ).
I Let c be a point just to the right of z .
I Let d be a point just to the right of y .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c
d
k = 3; colors: 1 2 3 4 5
I How should we color y?
I R[x ] canonical: y must get same color as z (color 2 ).
I Let c be a point just to the right of z .
I Let d be a point just to the right of y .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I How should we color y?
I R[x ] canonical: y must get same color as z (color 2 ).
I Let c be a point just to the right of z .
I Let d be a point just to the right of y .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
z
yyvv
x
z
yv
vy
x
zv
yy
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
zv
y
y
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
zv
x
zv
y
y
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
zv
x
zv
y
y
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Coloring Clean Families
x
zv
x
zv
y
y
c d
k = 3; colors: 1 2 3 4 5
I But z and v were inclusion-maximal.
I The coloring is canonical on X c and X d .
I At most 2 colors used on X c and at most 2 on X d .
I Some color is not used on X c or X d (color 5 ).
I Swap 2 and 5 on X>c .
I Now y can get color 2 .
Segments in Circle Graphs
I A center of X is the intersection of two overlapping intervals.
I A segment of X is an inclusion-maximal center.
LemmaIf X is a family of intervals whose overlap graph has clique numberk, then the overlap graph of segments of X has clique numberk − 1.
Segments in Circle Graphs
I A center of X is the intersection of two overlapping intervals.
I A segment of X is an inclusion-maximal center.
LemmaIf X is a family of intervals whose overlap graph has clique numberk, then the overlap graph of segments of X has clique numberk − 1.
Segments in Circle Graphs
I A center of X is the intersection of two overlapping intervals.
I A segment of X is an inclusion-maximal center.
LemmaIf X is a family of intervals whose overlap graph has clique numberk, then the overlap graph of segments of X has clique numberk − 1.
Segments in Circle Graphs
I A center of X is the intersection of two overlapping intervals.
I A segment of X is an inclusion-maximal center.
LemmaIf X is a family of intervals whose overlap graph has clique numberk, then the overlap graph of segments of X has clique numberk − 1.
Segments in Circle Graphs
I A center of X is the intersection of two overlapping intervals.
I A segment of X is an inclusion-maximal center.
LemmaIf X is a family of intervals whose overlap graph has clique numberk, then the overlap graph of segments of X has clique numberk − 1.
Segments in Circle Graphs
I Let X be a family of intervals.
I Let Y be the set of intervals contained in segments of X .
I Let Z = X − Y .
I Since Z is clean, we can color it with few colors.
I When the clique number is small, the segments are highlystructured.
I Using this and other tricks, we color Y .
Segments in Circle Graphs
I Let X be a family of intervals.
I Let Y be the set of intervals contained in segments of X .
I Let Z = X − Y .
I Since Z is clean, we can color it with few colors.
I When the clique number is small, the segments are highlystructured.
I Using this and other tricks, we color Y .
Segments in Circle Graphs
I Let X be a family of intervals.
I Let Y be the set of intervals contained in segments of X .
I Let Z = X − Y .
I Since Z is clean, we can color it with few colors.
I When the clique number is small, the segments are highlystructured.
I Using this and other tricks, we color Y .
Segments in Circle Graphs
I Let X be a family of intervals.
I Let Y be the set of intervals contained in segments of X .
I Let Z = X − Y .
I Since Z is clean, we can color it with few colors.
I When the clique number is small, the segments are highlystructured.
I Using this and other tricks, we color Y .
Segments in Circle Graphs
I Let X be a family of intervals.
I Let Y be the set of intervals contained in segments of X .
I Let Z = X − Y .
I Since Z is clean, we can color it with few colors.
I When the clique number is small, the segments are highlystructured.
I Using this and other tricks, we color Y .
Segments in Circle Graphs
I Let X be a family of intervals.
I Let Y be the set of intervals contained in segments of X .
I Let Z = X − Y .
I Since Z is clean, we can color it with few colors.
I When the clique number is small, the segments are highlystructured.
I Using this and other tricks, we color Y .
Chromatic Numbers of K4-free Circle Graphs
I When G is a K4-free circle graph, the Kostochka bound yieldsχ(G ) ≤ 120.
TheoremIf G is a K4-free circle graph, then χ(G ) ≤ 38.
I What is the maximum possible chromatic number of a K4-freecircle graph?
I No non-trivial lower bounds known.
I What is the maximum possible chromatic number of a cleancircle graph with clique number k?
Thank You.
Chromatic Numbers of K4-free Circle Graphs
I When G is a K4-free circle graph, the Kostochka bound yieldsχ(G ) ≤ 120.
TheoremIf G is a K4-free circle graph, then χ(G ) ≤ 38.
I What is the maximum possible chromatic number of a K4-freecircle graph?
I No non-trivial lower bounds known.
I What is the maximum possible chromatic number of a cleancircle graph with clique number k?
Thank You.
Chromatic Numbers of K4-free Circle Graphs
I When G is a K4-free circle graph, the Kostochka bound yieldsχ(G ) ≤ 120.
TheoremIf G is a K4-free circle graph, then χ(G ) ≤ 38.
I What is the maximum possible chromatic number of a K4-freecircle graph?
I No non-trivial lower bounds known.
I What is the maximum possible chromatic number of a cleancircle graph with clique number k?
Thank You.
Chromatic Numbers of K4-free Circle Graphs
I When G is a K4-free circle graph, the Kostochka bound yieldsχ(G ) ≤ 120.
TheoremIf G is a K4-free circle graph, then χ(G ) ≤ 38.
I What is the maximum possible chromatic number of a K4-freecircle graph?
I No non-trivial lower bounds known.
I What is the maximum possible chromatic number of a cleancircle graph with clique number k?
Thank You.
Chromatic Numbers of K4-free Circle Graphs
I When G is a K4-free circle graph, the Kostochka bound yieldsχ(G ) ≤ 120.
TheoremIf G is a K4-free circle graph, then χ(G ) ≤ 38.
I What is the maximum possible chromatic number of a K4-freecircle graph?
I No non-trivial lower bounds known.
I What is the maximum possible chromatic number of a cleancircle graph with clique number k?
Thank You.
Chromatic Numbers of K4-free Circle Graphs
I When G is a K4-free circle graph, the Kostochka bound yieldsχ(G ) ≤ 120.
TheoremIf G is a K4-free circle graph, then χ(G ) ≤ 38.
I What is the maximum possible chromatic number of a K4-freecircle graph?
I No non-trivial lower bounds known.
I What is the maximum possible chromatic number of a cleancircle graph with clique number k?
Thank You.