3-coloring triangle-free planar graphs with a precolored 9-cycle - PowerPoint PPT Presentation

3-coloring triangle-free planar graphs with a precolored 9-cycle 3-coloring triangle-free planar graphs with a precolored 9-cycle ILKYOO CHOI 1 , Jan Ekstein 2 , Pˇ remysl Holub 2 , Bernard Lidick´ y 1 University of Illinois at Urbana-Champaign, USA University of West Bohemia, Czech Republic December 21, 2013

3-coloring triangle-free planar graphs with a precolored 9-cycle A graph G is k -colorable if there is a function f where – for each vertex v : f ( v ) ∈ [ k ] – for each edge xy : f ( x ) � = f ( y ) A graph G is k -critical if – G is not ( k − 1)-colorable – for each subgraph H : H is ( k − 1)-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle A graph G is k -colorable if there is a function f where – for each vertex v : f ( v ) ∈ [ k ] – for each edge xy : f ( x ) � = f ( y ) A graph G is k -critical if – G is not ( k − 1)-colorable – for each subgraph H : H is ( k − 1)-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle A graph G is k -colorable if there is a function f where – for each vertex v : f ( v ) ∈ [ k ] – for each edge xy : f ( x ) � = f ( y ) A graph G is k -critical if – G is not ( k − 1)-colorable – for each subgraph H : H is ( k − 1)-colorable A graph G is C -critical for k -coloring if – for each edge e , there is a k -coloring f e of V ( C ) where – f e extends to G − e – f e does not extend to G

3-coloring triangle-free planar graphs with a precolored 9-cycle A graph G is k -colorable if there is a function f where – for each vertex v : f ( v ) ∈ [ k ] – for each edge xy : f ( x ) � = f ( y ) A graph G is k -critical if – G is not ( k − 1)-colorable – for each subgraph H : H is ( k − 1)-colorable A graph G is C -critical for k -coloring if – for each edge e , there is a k -coloring f e of V ( C ) where – f e extends to G − e – f e does not extend to G Observation If G is ( k + 1) -critical, then G is ∅ -critical for k-coloring.

3-coloring triangle-free planar graphs with a precolored 9-cycle 4-critical – not 3-colorable – each subgraph is 3-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle G 1 G 2 e C 4-critical – not 3-colorable – each subgraph is 3-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle G 1 G 2 e C 4-critical – not 3-colorable – each subgraph is 3-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle 2 G 1 G 2 1 1 1 3 e 3 2 C 4-critical – not 3-colorable – each subgraph is 3-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle Observation There exists a 3 -coloring of V ( C ) that extends to G 1 − e but does not extend to G 1 . 2 G 1 G 2 1 1 1 3 e 3 2 C 4-critical – not 3-colorable – each subgraph is 3-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle Observation There exists a 3 -coloring of V ( C ) that extends to G 1 − e but does not extend to G 1 . 2 G 1 G 2 e 2 C 4-critical – not 3-colorable – each subgraph is 3-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle Observation There exists a 3 -coloring of V ( C ) that extends to G 1 − e but does not extend to G 1 . Observation For every cut C and every e ∈ V ( G 1 ) exists a 3 -coloring of V ( C ) that extends to G 1 − e but does not extend to G 1 . 2 G 1 G 2 e 2 C 4-critical – not 3-colorable – each subgraph is 3-colorable

3-coloring triangle-free planar graphs with a precolored 9-cycle Definition A graph G is C-critical for k -coloring if for each e ∈ E ( G ), there exists a k -coloring f e of V ( C ) that extends to G − e but does not extend to G .

3-coloring triangle-free planar graphs with a precolored 9-cycle Definition A graph G is C-critical for k -coloring if for each e ∈ E ( G ), there exists a k -coloring f e of V ( C ) that extends to G − e but does not extend to G . e 1 e 2 C

3-coloring triangle-free planar graphs with a precolored 9-cycle Definition A graph G is C-critical for k -coloring if for each e ∈ E ( G ), there exists a k -coloring f e of V ( C ) that extends to G − e but does not extend to G . e 1 e 2 C 1 2 1 2 e 1 e 2 C C ϕ 1 ϕ 2 1 2 3 1 3 3 2 3 1 2

3-coloring triangle-free planar graphs with a precolored 9-cycle Definition A graph G is C-critical for k -coloring if for each e ∈ E ( G ), there exists a k -coloring f e of V ( C ) that extends to G − e but does not extend to G . e 1 e 2 C 1 2 1 2 e 1 e 2 C C ϕ 1 ϕ 2 1 2 3 1 3 3 2 3 1 2 Observation If G is ( k + 1) -critical, then G is ∅ -critical for k-coloring.

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice?

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces G G 1 G 2 ⇒ +

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces G 2 G G 1 G 2 ⇒ +

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces G 2 G G 1 G 2 ⇒ + precolored tree

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces G 2 G G 1 G 2 ⇒ + precolored tree G

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces G 2 G G 1 G 2 ⇒ + precolored tree G interior of a cycle

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces G 2 G G 1 G 2 ⇒ + precolored tree G interior of a cycle G 1 G 2 G

3-coloring triangle-free planar graphs with a precolored 9-cycle – Why C -critical? Which C is a good choice? simplifying graphs on surfaces G 2 G 2 G G 1 G 2 ⇒ + precolored tree G G interior of a cycle G 2 G 1 G 2 G

3-coloring triangle-free planar graphs with a precolored 9-cycle Theorem (Gr¨ otzsch 1959, Aksenov 1974) If G is a plane graph of girth 4 , then a pre-coloring of either a 4 -cycle or a 5 -cycle extends to 3 -coloring of G.

3-coloring triangle-free planar graphs with a precolored 9-cycle Theorem (Gr¨ otzsch 1959, Aksenov 1974) If G is a plane graph of girth 4 , then a pre-coloring of either a 4 -cycle or a 5 -cycle extends to 3 -coloring of G. Focus: plane graphs that are C -critical for 3-coloring where C is a cycle.

3-coloring triangle-free planar graphs with a precolored 9-cycle Theorem (Gr¨ otzsch 1959, Aksenov 1974) If G is a plane graph of girth 4 , then a pre-coloring of either a 4 -cycle or a 5 -cycle extends to 3 -coloring of G. Focus: plane graphs that are C -critical for 3-coloring where C is a cycle. Goal: Characterize all C -critical plane graphs of girth 4.

3-coloring triangle-free planar graphs with a precolored 9-cycle Theorem (Gr¨ otzsch 1959, Aksenov 1974) If G is a plane graph of girth 4 , then a pre-coloring of either a 4 -cycle or a 5 -cycle extends to 3 -coloring of G. Focus: plane graphs that are C -critical for 3-coloring where C is a cycle. Goal: Characterize all C -critical plane graphs of girth 4. STILL OPEN!

3-coloring triangle-free planar graphs with a precolored 9-cycle Theorem (Gr¨ otzsch 1959, Aksenov 1974) If G is a plane graph of girth 4 , then a pre-coloring of either a 4 -cycle or a 5 -cycle extends to 3 -coloring of G. Focus: plane graphs that are C -critical for 3-coloring where C is a cycle. Goal: Characterize all C -critical plane graphs of girth 4. STILL OPEN! Easier goal: Characterize all C -critical plane graphs of girth 5.

3-coloring triangle-free planar graphs with a precolored 9-cycle Theorem (Gr¨ otzsch 1959, Aksenov 1974) If G is a plane graph of girth 4 , then a pre-coloring of either a 4 -cycle or a 5 -cycle extends to 3 -coloring of G. Focus: plane graphs that are C -critical for 3-coloring where C is a cycle. Goal: Characterize all C -critical plane graphs of girth 4. STILL OPEN! Easier goal: Characterize all C -critical plane graphs of girth 5. SOLVED!

3-coloring triangle-free planar graphs with a precolored 9-cycle Theorem (Gr¨ otzsch 1959, Aksenov 1974) If G is a plane graph of girth 4 , then a pre-coloring of either a 4 -cycle or a 5 -cycle extends to 3 -coloring of G. Focus: plane graphs that are C -critical for 3-coloring where C is a cycle. Goal: Characterize all C -critical plane graphs of girth 4. STILL OPEN! Easier goal: Characterize all C -critical plane graphs of girth 5. SOLVED! | C | ≤ 11 by Thomassen 2003 and Walls 1999 | C | = 12 by Dvoˇ r´ ak–Kawarabayashi 2011