[PPT] - Extending 3-coloring of a face in triangle-free planar graphs Zden PowerPoint Presentation

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Extending 3-coloring of a face in triangle-free planar graphs

Zdenˇ ek Dvoˇ rák, Bernard Lidický

Charles University in Prague University of Illinois at Urbana-Champaign

AMS Sectional - Louisville October 6, 2013

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Definitions (critical graphs)

graph G = (V, E) coloring is ϕ : V → K such that ϕ(u) = ϕ(v) if uv ∈ E G is a k-colorable if coloring with |K| = k exists G is a k-critical graph if G is not (k − 1)-colorable but every H ⊂ G is (k − 1)-colorable.

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Cuts in a 4-critical graph G

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Cuts in a 4-critical graph G

e C G1 G2

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Cuts in a 4-critical graph G

e C G1 G2

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Cuts in a 4-critical graph G

e C G1 G2 1 1 2 3 2 1 3

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Cuts in a 4-critical graph G

Observation

There exists a 3-coloring of V(C) that extends to G1 − e but does not extend to G1.

e C G1 G2 2 2

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Cuts in a 4-critical graph G

Observation

For every cut C and every e ∈ V(G1) exists a 3-coloring of V(C) that extends to G1 − e but does not extend to G1.

e C G1 G2 2 2

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Different colorings for different edges

e1 e2 C

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Different colorings for different edges

1 2 3 1 3 1 2 e1 C ϕ1 3 2 3 1 2 1 2 e2 C ϕ2

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Different colorings for different edges

Definition

A graph G is C-critical for k-coloring if for every e ∈ E(G) exists a k-coloring ϕe of V(C) that extends to G − e but does not extend to G.

1 2 3 1 3 1 2 e1 C ϕ1 3 2 3 1 2 1 2 e2 C ϕ2

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Different colorings for different edges

Definition

A graph G is C-critical for k-coloring if for every e ∈ E(G) exists a k-coloring ϕe of V(C) that extends to G − e but does not extend to G.

Observation

If G is (k + 1)-critical, then G is ∅-critical for k-coloring.

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Which C for C-critical?

simplifying graphs on surfaces

G

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Which C for C-critical?

simplifying graphs on surfaces

G G1 G2

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Which C for C-critical?

simplifying graphs on surfaces

G G1 G2

G2

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Which C for C-critical?

simplifying graphs on surfaces

G2

interior of a cycle

G2 G1 G

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Which C for C-critical?

simplifying graphs on surfaces

G2

interior of a cycle

G2 G1 G

G2

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Which C for C-critical?

simplifying graphs on surfaces

G2

interior of a cycle

G2

precolored tree

G

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Which C for C-critical?

simplifying graphs on surfaces

G2 G2

interior of a cycle

G2

precolored tree

G

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Our focus

We focus on G that is

plane
outer-face is a cycle C
G is C-critical for 3-coloring

Goal: For a given length of C enumerate all C-critical graphs.

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Our focus

We focus on G that is

plane
outer-face is a cycle C
G is C-critical for 3-coloring

Goal: For a given length of C enumerate all C-critical graphs.

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Known results (girth 5)

C-critical plane graphs of girth 5 are precisely enumerated for

|C| ≤ 11 by Thomassen ’03 and Walls ’99
|C| = 12 by Dvoˇ

rák and Kawarabayashi ’11

|C| ≤ 16 by Dvoˇ

rák and L. ’13+

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Known results (girth 5)

C-critical plane graphs of girth 5 are precisely enumerated for

|C| ≤ 11 by Thomassen ’03 and Walls ’99
|C| = 12 by Dvoˇ

rák and Kawarabayashi ’11

|C| ≤ 16 by Dvoˇ

rák and L. ’13+ Recursive description for all |C| by Dvoˇ rák and Kawarabayashi ’11

(a) (b) (c) (d)

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|C| ≤ 10 (girth 5)

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Known results (girth 4)

No recursive enumeration for girth 4 know. C-critical plane graphs of girth 4 precisely enumerated for

|C| ∈ {4, 5} by Aksenov ’74
|C| = 6 by Gimbel and Thomassen ’97
|C| = 6 by Aksenov, Borodin, and Glebov ’03
|C| = 7 by Aksenov, Borodin, and Glebov ’04
|C| = 8 by Dvoˇ

rák and L. ’13+

|C| = 9 by Choi, Ekstein, Holub, and L. (in writing)

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|C| ∈ {4, 5, 6} (girth 4)

Theorem (Aksenov ’74)

If G is a plane graph of girth 4, then every pre-coloring of C4 and C5 extends to G.

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|C| ∈ {4, 5, 6} (girth 4)

Theorem (Aksenov ’74)

If G is a plane graph of girth 4, then every pre-coloring of C4 and C5 extends to G.

Theorem (Gimbel and Thomassen ’97; Aksenov, Borodin, and Glebov ’03)

Let G be a plane triangle-free graph with chordless outer 6-cycle C. G is C-critical if and only if G contains no separating 4-cycles and all other faces of G are 4-faces (i.e. G is a quadrangulation). Moreover, a 3-coloring of C does not extend to G if and only if opposite vertices of C are colored the same.

1 2 3 1 2 3

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|C| = 7 (girth 4)

Theorem (Aksenov, Borodin, and Glebov ’04)

If G is a plane triangle-free graph with outer face bounded by a cycle C of length 7 then G is C-critical iff G looks like (a), (b), or (c).

1 1 (a) 2 1 3 2 1 (b) 1 3 2 1 2 3 2 (c)

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|C| = 8 (girth 4)

Theorem (Dvoˇ rák and L.)

If G is a plane triangle-free graph with outer face bounded by a cycle C of length 8 then G is C-critical iff G looks like (a), (b), (c), or (d).

1 3 2 1 3 2 1 2 (a) 1 3 2 1 3 2 1 2 (a) 1 2 1 3 2 1 3 2 (b) 1 2 1 3 2 1 3 2 (b) 2 3 2 1 2 3 2 1 (c) 2 3 2 1 2 3 2 1 (c) 1 2 3 2 1 3 2 3 (d) 1 2 3 2 1 3 2 3 (d)

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Tool

Theorem (Tutte ’54)

A plane graph G has a 3-coloring iff its dual G⋆ has a nowhere-zero Z3-flow.

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Tool

Theorem (Tutte ’54)

A plane graph G has a 3-coloring iff its dual G⋆ has a nowhere-zero Z3-flow.

1 2 2 3 3 1

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Tool

Theorem (Tutte ’54)

A plane graph G has a 3-coloring iff its dual G⋆ has a nowhere-zero Z3-flow.

1 2 2 3 3 1

2 1 3 3 1 2 3 2 1 2 1 2 1 2

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C-critical quadrangulation

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C-critical quadrangulation

1 3 2 1 3 2

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C-critical quadrangulation

1 3 2 1 3 2

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C-critical quadrangulation

1 3 2 1 3 2 3 2 1 2 1 2 1 2

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C-critical quadrangulation

e

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C-critical quadrangulation

1 3 2 1 3 2 e

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C-critical quadrangulation

1 3 2 1 3 2 e

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C-critical quadrangulation

1 3 2 1 3 2 e

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C-critical quadrangulation

1 3 2 1 3 2 e

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C-critical quadrangulation

1 3 2 1 3 2 e

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|C| = 8

Corollary (Dvoˇ rák, Král’, Thomas)

If G is a C-critical, plane, triangle-free graph, where |C| = 8, then {∅, {7}, {5, 5}} are the only possible multisets of face lengths ≥ 5.

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|C| = 8

Corollary (Dvoˇ rák, Král’, Thomas)

If G is a C-critical, plane, triangle-free graph, where |C| = 8, then {∅, {7}, {5, 5}} are the only possible multisets of face lengths ≥ 5.

3 2 1 2 1 2 2 1 2 1

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

3 2 1 2 1 2 2 1 2 1

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

3 2 1 2 1 2 2 1 2 1

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

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|C| = 8, two 5-faces

Finding G and a coloring of C that does not extend. "source edges = sink edges"

s t

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