3-coloring graphs without induced paths on seven vertices Oliver - PowerPoint PPT Presentation

3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit¨ at zu K¨ oln, Institut f¨ ur Informatik with Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Maya Stein, and Mingxian Zhong

Graph coloring

Graph coloring ◮ a proper k-coloring is an assignment of numbers { 1 , 2 , . . . , k } to the vertices such that any two adjacent vertices receive distinct numbers

Graph coloring ◮ a proper k-coloring is an assignment of numbers { 1 , 2 , . . . , k } to the vertices such that any two adjacent vertices receive distinct numbers

Graph coloring ◮ a proper k-coloring is an assignment of numbers { 1 , 2 , . . . , k } to the vertices such that any two adjacent vertices receive distinct numbers 4 1 2 3 1 4 2 1 2 1 3

Graph coloring ◮ a proper k-coloring is an assignment of numbers { 1 , 2 , . . . , k } to the vertices such that any two adjacent vertices receive distinct numbers 4 1 2 3 1 4 2 1 2 1 3 ◮ the related decision problem is called k-colorability

Graph coloring ◮ a proper k-coloring is an assignment of numbers { 1 , 2 , . . . , k } to the vertices such that any two adjacent vertices receive distinct numbers 4 1 2 3 1 4 2 1 2 1 3 ◮ the related decision problem is called k-colorability ◮ it is NP-complete for every k ≥ 3

k -colorability in H -free graphs

k -colorability in H -free graphs ◮ fix some graph H

k -colorability in H -free graphs ◮ fix some graph H ◮ a graph is H-free if it does not contain H as an induced subgraph

k -colorability in H -free graphs ◮ fix some graph H ◮ a graph is H-free if it does not contain H as an induced subgraph Theorem (Lozin and Kaminski, 2007) 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 .

k -colorability in P t -free graphs

k -colorability in P t -free graphs ◮ P t is the path on t vertices

k -colorability in P t -free graphs ◮ P t is the path on t vertices ◮ complexity of k -colorability in the class of P t -free graphs:

k -colorability in P t -free graphs ◮ P t is the path on t vertices ◮ complexity of k -colorability in the class of P t -free graphs: k/t 4 5 6 7 8 9 . . . 3 O ( m ) O ( n α ) O ( mn α ) P ∗ ? ? . . . 4 O ( m ) P ? NPC NPC NPC . . . 5 O ( m ) P NPC NPC NPC NPC . . . 6 O ( m ) P NPC NPC NPC NPC . . . . . . . . . . ... . . . . . . . . . . . . . .

k -colorability in P t -free graphs ◮ P t is the path on t vertices ◮ complexity of k -colorability in the class of P t -free graphs: k/t 4 5 6 7 8 9 . . . 3 O ( m ) O ( n α ) O ( mn α ) P ∗ ? ? . . . 4 O ( m ) P ? NPC NPC NPC . . . 5 O ( m ) P NPC NPC NPC NPC . . . 6 O ( m ) P NPC NPC NPC NPC . . . . . . . . . . ... . . . . . . . . . . . . . . ◮ *) our contribution

3-colorability in P 7 -free graphs

3-colorability in P 7 -free graphs ◮ algorithm is split into two cases

3-colorability in P 7 -free graphs ◮ algorithm is split into two cases ◮ input graphs with or without a triangle

3-colorability in P 7 -free graphs ◮ algorithm is split into two cases ◮ input graphs with or without a triangle

3-colorability in P 7 -free graphs ◮ algorithm is split into two cases ◮ input graphs with or without a triangle

3-colorability in P 7 -free graphs ◮ algorithm is split into two cases ◮ input graphs with or without a triangle ◮ triangle-free case is on arxiv (since September)

3-colorability in P 7 -free graphs ◮ algorithm is split into two cases ◮ input graphs with or without a triangle ◮ triangle-free case is on arxiv (since September) ◮ case of graphs contaning a triangle is in preparation

Triangle-free case

Triangle-free case Theorem (BCMSSZ, 2014) Given a ( P 7 , triangle ) -free graph G, it can be decided in O ( | V ( G ) | 7 ) time whether G admits a 3-coloring.

Triangle-free case Theorem (BCMSSZ, 2014) Given a ( P 7 , triangle ) -free graph G, it can be decided in O ( | V ( G ) | 7 ) time whether G admits a 3-coloring. ◮ algorithm based on a mild structural analysis

Triangle-free case Theorem (BCMSSZ, 2014) Given a ( P 7 , triangle ) -free graph G, it can be decided in O ( | V ( G ) | 7 ) time whether G admits a 3-coloring. ◮ algorithm based on a mild structural analysis & smart enumeration

Triangle-free case Theorem (BCMSSZ, 2014) Given a ( P 7 , triangle ) -free graph G, it can be decided in O ( | V ( G ) | 7 ) time whether G admits a 3-coloring. ◮ algorithm based on a mild structural analysis & smart enumeration & reduction to 2-SAT

Triangle-free case Theorem (BCMSSZ, 2014) Given a ( P 7 , triangle ) -free graph G, it can be decided in O ( | V ( G ) | 7 ) time whether G admits a 3-coloring. ◮ algorithm based on a mild structural analysis & smart enumeration & reduction to 2-SAT ◮ it actually solves the list 3-colorability problem

Triangle-free case Theorem (BCMSSZ, 2014) Given a ( P 7 , triangle ) -free graph G, it can be decided in O ( | V ( G ) | 7 ) time whether G admits a 3-coloring. ◮ algorithm based on a mild structural analysis & smart enumeration & reduction to 2-SAT ◮ it actually solves the list 3-colorability problem ◮ every vertex is equipped with a subset of { 1 , 2 , 3 } of admissible colors

◮ first we determine the core structure of the input graph:

◮ first we determine the core structure of the input graph:

◮ outside its core structure, the graph is bipartite

◮ the non-trival components outside the core structure are well-behaved

◮ the non-trival components outside the core structure are well-behaved C 1 C 2

◮ the non-trival components outside the core structure are well-behaved T i C 1 C 2

◮ the non-trival components outside the core structure are well-behaved T i C 1 N ( C 1 ) C 2

◮ the non-trival components outside the core structure are well-behaved T i C 1 N ( C 1 ) N ( C 2 ) C 2

◮ the non-trival components outside the core structure are well-behaved T i C 1 N ( C 1 ) N ( C 2 ) . C 2 . .

Triangle-free case

Triangle-free case ◮ first we enumerate some partial colorings of the sets T i

Triangle-free case ◮ first we enumerate some partial colorings of the sets T i ◮ determined by the nested neighborhoods of the non-trivial components

Triangle-free case ◮ first we enumerate some partial colorings of the sets T i ◮ determined by the nested neighborhoods of the non-trivial components ◮ leaves a 2-SAT problem for the coloring of the non-trivial components outside the core structure

Triangle-free case ◮ first we enumerate some partial colorings of the sets T i ◮ determined by the nested neighborhoods of the non-trivial components ◮ leaves a 2-SAT problem for the coloring of the non-trivial components outside the core structure ◮ then we enumerate some partial colorings of the sets D j

Triangle-free case ◮ first we enumerate some partial colorings of the sets T i ◮ determined by the nested neighborhoods of the non-trivial components ◮ leaves a 2-SAT problem for the coloring of the non-trivial components outside the core structure ◮ then we enumerate some partial colorings of the sets D j ◮ leaves a 2-SAT problem for the trivial components

Triangle-free case ◮ first we enumerate some partial colorings of the sets T i ◮ determined by the nested neighborhoods of the non-trivial components ◮ leaves a 2-SAT problem for the coloring of the non-trivial components outside the core structure ◮ then we enumerate some partial colorings of the sets D j ◮ leaves a 2-SAT problem for the trivial components ◮ then we color the rest of the core structure

Open problems

Open problems ◮ Is there a t such that 3-colorability is NP-complete in P t -free graphs? ◮ Is k -colorability FPT in the class of P 5 -free graphs? ◮ Are there finitely many 4-vertex-critical P 6 -free graphs?

Open problems ◮ Is there a t such that 3-colorability is NP-complete in P t -free graphs? ◮ Is k -colorability FPT in the class of P 5 -free graphs? ◮ Are there finitely many 4-vertex-critical P 6 -free graphs? Thanks!