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Obstructions against 3-coloring graphs without long induced paths
Oliver Schaudt
Universit¨ at zu K¨
- ln & RWTH Aachen
Obstructions against 3-coloring graphs without long induced paths - - PowerPoint PPT Presentation
Obstructions against 3-coloring graphs without long induced paths - - PowerPoint PPT Presentation
Obstructions against 3-coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH Aachen with Maria Chudnovsky, Jan Goedgebeur, and Mingxian Zhong Graph coloring Graph coloring a k-coloring is an
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Graph coloring
◮ a k-coloring is an assignment of numbers {1, 2, . . . , k} to the
vertices such that any two adjacent vertices receive distinct numbers
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Graph coloring
◮ a k-coloring is an assignment of numbers {1, 2, . . . , k} to the
vertices such that any two adjacent vertices receive distinct numbers
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Graph coloring
◮ a k-coloring is an assignment of numbers {1, 2, . . . , k} to the
vertices such that any two adjacent vertices receive distinct numbers 1 1 1 1 2 2 2 3 3 3 4
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Graph coloring
◮ a k-coloring is an assignment of numbers {1, 2, . . . , k} to the
vertices such that any two adjacent vertices receive distinct numbers 1 1 1 1 2 2 2 3 3 3 4
◮ the related decision problem is called k-colorability
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Graph coloring
◮ a k-coloring is an assignment of numbers {1, 2, . . . , k} to the
vertices such that any two adjacent vertices receive distinct numbers 1 1 1 1 2 2 2 3 3 3 4
◮ the related decision problem is called k-colorability ◮ it is NP-complete for every k ≥ 3
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k-colorability in H-free graphs
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k-colorability in H-free graphs
◮ fix some graph H
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k-colorability in H-free graphs
◮ fix some graph H ◮ a graph G is H-free if it does not contain H as an induced
subgraph
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k-colorability in H-free graphs
◮ fix some graph H ◮ a graph G is H-free if it does not contain H as an induced
subgraph
Theorem (Lozin and Kaminski 2007, Kr´ al et al. 2001)
Let H be any graph that is not the disjoint union of paths. Then k-colorability is NP-complete in the class of H-free graphs, for all k ≥ 3.
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k-colorability in H-free graphs
◮ fix some graph H ◮ a graph G is H-free if it does not contain H as an induced
subgraph
Theorem (Lozin and Kaminski 2007, Kr´ al et al. 2001)
Let H be any graph that is not the disjoint union of paths. Then k-colorability is NP-complete in the class of H-free graphs, for all k ≥ 3.
◮ leads to the study of Pt-free graphs
1 2 3 t · · · · · ·
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3-colorability in Pt-free graphs
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3-colorability in Pt-free graphs
Theorem (Randerath and Schiermeyer 2004)
The 3-colorability problem can be solved in polynomial time for P6-free graphs.
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3-colorability in Pt-free graphs
Theorem (Randerath and Schiermeyer 2004)
The 3-colorability problem can be solved in polynomial time for P6-free graphs.
Theorem (Bonomo, Chudnovsky, Maceli, S, Stein, Zhong ’14)
The 3-colorability problem can be solved in polynomial time for P7-free graphs.
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3-colorability in Pt-free graphs
Theorem (Randerath and Schiermeyer 2004)
The 3-colorability problem can be solved in polynomial time for P6-free graphs.
Theorem (Bonomo, Chudnovsky, Maceli, S, Stein, Zhong ’14)
The 3-colorability problem can be solved in polynomial time for P7-free graphs.
Open Problem
Is there any t such that 3-colorability is NP-hard for Pt-free graphs?
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Obstructions against 3-colorability
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Obstructions against 3-colorability
1 1 1 1 2 2 2 3 3 3 4
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Obstructions against 3-colorability
1 1 1 1 2 2 2 3 3 3 4
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Obstructions against 3-colorability
1 1 1 1 2 2 2 3 3 3 4
◮ 4-critical graph: needs four colors, but every proper subgraph
is 3-colorable
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Obstructions against 3-colorability
1 1 1 1 2 2 2 3 3 3 4
◮ 4-critical graph: needs four colors, but every proper subgraph
is 3-colorable
◮ call such a graph an obstruction against 3-colorability
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Obstructions against 3-colorability
1 1 1 1 2 2 2 3 3 3 4
◮ 4-critical graph: needs four colors, but every proper subgraph
is 3-colorable
◮ call such a graph an obstruction against 3-colorability ◮ useful in the design of certifying algorithms
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Obstructions against 3-colorability
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Obstructions against 3-colorability
Theorem (Randerath, Schiermeyer and Tewes 2002)
The only obstruction in the class of (P6, K3)-free graphs is the Gr¨
- tzsch graph.
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Obstructions against 3-colorability
Theorem (Randerath, Schiermeyer and Tewes 2002)
The only obstruction in the class of (P6, K3)-free graphs is the Gr¨
- tzsch graph.
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Obstructions against 3-colorability
Theorem (Randerath, Schiermeyer and Tewes 2002)
The only obstruction in the class of (P6, K3)-free graphs is the Gr¨
- tzsch graph.
Theorem (Bruce, H`
- ang and Sawada 2009)
There are six obstructions in the class of P5-free graphs.
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Obstructions against 3-colorability
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Obstructions against 3-colorability
◮ Golovach et al.: is there a certifying algorithm for
3-colorability on P6-free graphs?
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Obstructions against 3-colorability
◮ Golovach et al.: is there a certifying algorithm for
3-colorability on P6-free graphs?
◮ Seymour: for which connected graphs H exist only finitely
many obstructions in the class of H-free graphs?
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Obstructions against 3-colorability
◮ Golovach et al.: is there a certifying algorithm for
3-colorability on P6-free graphs?
◮ Seymour: for which connected graphs H exist only finitely
many obstructions in the class of H-free graphs?
Theorem (Chudnovsky, Goedgebeur, S and Zhong 2015)
There are 24 obstructions in the class of P6-free graphs.
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Obstructions against 3-colorability
◮ Golovach et al.: is there a certifying algorithm for
3-colorability on P6-free graphs?
◮ Seymour: for which connected graphs H exist only finitely
many obstructions in the class of H-free graphs?
Theorem (Chudnovsky, Goedgebeur, S and Zhong 2015)
There are 24 obstructions in the class of P6-free graphs. Moreover, if H is connected and not a subgraph of P6, there are infinitely many obstructions in the class of H-free graphs.
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Structure of the proof
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Structure of the proof
◮ Prove that our list is complete in the (P6, diamond)-free case
◮ Use an automatic proof, building on a method of H`
- ang et al.
◮ Exhaustive enumeration, exploiting properties of obstructions
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Structure of the proof
◮ Prove that our list is complete in the (P6, diamond)-free case
◮ Use an automatic proof, building on a method of H`
- ang et al.
◮ Exhaustive enumeration, exploiting properties of obstructions
◮ Prove that our list is complete up to 28 vertices
◮ Use same enumeration algorithm
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Structure of the proof
◮ Prove that our list is complete in the (P6, diamond)-free case
◮ Use an automatic proof, building on a method of H`
- ang et al.
◮ Exhaustive enumeration, exploiting properties of obstructions
◮ Prove that our list is complete up to 28 vertices
◮ Use same enumeration algorithm
◮ Prove that our list is complete in the full case
◮ Structural analysis by hand ◮ Contraction/Decontraction of maximal tripods
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Obstructions against 3-colorability
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Obstructions against 3-colorability
◮ There is an infinite family of P7-free obstructions
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Obstructions against 3-colorability
◮ There is an infinite family of P7-free obstructions
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Obstructions against 3-colorability
◮ There is an infinite family of P7-free obstructions ◮ Easy: infinte familes of claw-free obstructions, and
- bstructions of large girth
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Obstructions against 3-colorability
◮ There is an infinite family of P7-free obstructions ◮ Easy: infinte familes of claw-free obstructions, and
- bstructions of large girth
◮ this yields the desired dichotomy
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Open problems
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Open problems
◮ Formulate a dichotomy theorem for general H ◮ Is 3-colorability solvable in polytime on Pt-free graphs? ◮ Is 4-colorability solvable in polytime on P6-free graphs? ◮ Is k-colorability FPT in the class of P5-free graphs?
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