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DP-coloring of planar graphs Runrun Liu Central China Normal University May 17, 2019 Runrun Liu DP-coloring of planar graphs Definitions: proper coloring and list coloring A (proper) k-coloring of G is a mapping f : V ( G ) [ k ] such that

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DP-coloring of planar graphs

Runrun Liu

Central China Normal University

May 17, 2019

Runrun Liu DP-coloring of planar graphs

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Definitions: proper coloring and list coloring

A (proper) k-coloring of G is a mapping f : V (G) → [k] such that f (u) = f (v) whenever uv ∈ E(G). The chromatic number χ(G) is the smallest k such that G has a k-coloring.

Runrun Liu DP-coloring of planar graphs

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Definitions: proper coloring and list coloring

A (proper) k-coloring of G is a mapping f : V (G) → [k] such that f (u) = f (v) whenever uv ∈ E(G). The chromatic number χ(G) is the smallest k such that G has a k-coloring.

Runrun Liu DP-coloring of planar graphs

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Definitions: proper coloring and list coloring

A list assignment of G is a function L that gives each vertex L(v) colors. A graph G is list colorable if there is a proper coloring c of V (G) such that c(v) ∈ L(v) for each v ∈ V (G).

Runrun Liu DP-coloring of planar graphs

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Definitions: proper coloring and list coloring

A list assignment of G is a function L that gives each vertex L(v) colors. A graph G is list colorable if there is a proper coloring c of V (G) such that c(v) ∈ L(v) for each v ∈ V (G). A graph G is k-choosable if G is list colorable for each L with |L(v)| ≥ k. The list chromatic number χℓ(G) is the smallest k such that G is k-choosable.

Runrun Liu DP-coloring of planar graphs

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Definitions: proper coloring and list coloring

A list assignment of G is a function L that gives each vertex L(v) colors. A graph G is list colorable if there is a proper coloring c of V (G) such that c(v) ∈ L(v) for each v ∈ V (G). A graph G is k-choosable if G is list colorable for each L with |L(v)| ≥ k. The list chromatic number χℓ(G) is the smallest k such that G is k-choosable. an example for that: χ(G) < χℓ(G).

Runrun Liu DP-coloring of planar graphs

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Definitions: proper coloring and list coloring

A list assignment of G is a function L that gives each vertex L(v) colors. A graph G is list colorable if there is a proper coloring c of V (G) such that c(v) ∈ L(v) for each v ∈ V (G). A graph G is k-choosable if G is list colorable for each L with |L(v)| ≥ k. The list chromatic number χℓ(G) is the smallest k such that G is k-choosable. an example for that: χ(G) < χℓ(G).

Runrun Liu DP-coloring of planar graphs

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Background of DP-coloring

All planar graphs without 4-cycles and 5-cycles are 3-colorable.

Erd 

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Find k, such that all planar graphs without cycles of length from 4 to k are 3-colorable. All planar graphs without cycles of length from 4 to 9 are 3-choosable. All planar graphs without cycles of length from 4 to 8 are 3-choosable.

Runrun Liu DP-coloring of planar graphs

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Background of DP-coloring

All planar graphs without 4-cycles and 5-cycles are 3-colorable.

Erd 

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Find k, such that all planar graphs without cycles of length from 4 to k are 3-colorable. All planar graphs without cycles of length from 4 to 9 are 3-choosable. All planar graphs without cycles of length from 4 to 8 are 3-choosable.

Theorem (Dvoˇ r´ ak and Postle, 2017) Every planar graph without cycles of length from 4 to 8 is 3-choosable.

Runrun Liu DP-coloring of planar graphs

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The definition of DP-coloring

In order to understand the definition easier, let’s consider another perspective of proper coloring and list coloring.

Runrun Liu DP-coloring of planar graphs

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The definition of DP-coloring

In order to understand the definition easier, let’s consider another perspective of proper coloring and list coloring. Example: Goal: find an independent set of order 4 (that intersects each color list exactly once)

Runrun Liu DP-coloring of planar graphs

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The definition of DP-coloring

In order to understand the definition easier, let’s consider another perspective of proper coloring and list coloring. Example: Goal: find an independent set of order 4 (that intersects each color list exactly once)

Runrun Liu DP-coloring of planar graphs

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The definition of DP-coloring

G

Graph G Host graph GL v ∈ V (G) a clique of |Lv| vertices uv ∈ E(G) a matching between Lu and Lv G is DP-k-colorable For each v ∈ V (G) with |Lv| ≥ k, there exists an independent set of |V (G)| vertices in GL for every matching assignment.

Runrun Liu DP-coloring of planar graphs

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The advantage of DP-coloring

In DP-coloring, it does not matter if we set all list assignments to be the same.

Runrun Liu DP-coloring of planar graphs

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The advantage of DP-coloring

In DP-coloring, it does not matter if we set all list assignments to be the same. So, one can impose constraints (“consistency”) on matching assignment so that locally it becomes proper coloring. So, identifying vertices becomes valid.

Runrun Liu DP-coloring of planar graphs

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The advantage of DP-coloring

In DP-coloring, it does not matter if we set all list assignments to be the same. So, one can impose constraints (“consistency”) on matching assignment so that locally it becomes proper coloring. So, identifying vertices becomes valid.

Runrun Liu DP-coloring of planar graphs

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If one can reduce the following structure, then Borodin’s conjecture would be confirmed.

Runrun Liu DP-coloring of planar graphs

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If one can reduce the following structure, then Borodin’s conjecture would be confirmed.

Runrun Liu DP-coloring of planar graphs

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Results 1

The result of Dvoˇ r´ ak and Postle can be strengthened:

Runrun Liu DP-coloring of planar graphs

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Results 1

The result of Dvoˇ r´ ak and Postle can be strengthened: Theorem (L.-Li, 2018+) Planar graphs without adjacent cycles of length at most 8 are 3-choosable.

Runrun Liu DP-coloring of planar graphs

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Results 1

The result of Dvoˇ r´ ak and Postle can be strengthened: Theorem (L.-Li, 2018+) Planar graphs without adjacent cycles of length at most 8 are 3-choosable. New results on 3-choosability can be proved:

Runrun Liu DP-coloring of planar graphs

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Results 1

The result of Dvoˇ r´ ak and Postle can be strengthened: Theorem (L.-Li, 2018+) Planar graphs without adjacent cycles of length at most 8 are 3-choosable. New results on 3-choosability can be proved: Theorem (Yin-Yu, 2018+) The following planar graphs are DP-3-colorable:

without {4, 5}-cycles and distance of triangles at least 3. without {4, 5, 6}-cycles and distance of triangles at least 2.

Runrun Liu DP-coloring of planar graphs

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Results 2

Some results on list 3-coloring can be strengthened:

Runrun Liu DP-coloring of planar graphs

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Results 2

Some results on list 3-coloring can be strengthened: Theorem (L.-Loeb-Yin-Yu, 2019) Planar graphs without {3, 6, 7, 8}-cycles, or {3, 5, 6}-cycles or {5, 6, 7}-cycles, or {4, 5, 6, 9}-cycles, or {4, 5, 7, 9}-cycles are DP-3-colorable.

Runrun Liu DP-coloring of planar graphs

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Results 2

Some results on list 3-coloring can be strengthened: Theorem (L.-Loeb-Yin-Yu, 2019) Planar graphs without {3, 6, 7, 8}-cycles, or {3, 5, 6}-cycles or {5, 6, 7}-cycles, or {4, 5, 6, 9}-cycles, or {4, 5, 7, 9}-cycles are DP-3-colorable. In particular, planar graphs without {4, a, b, 9}-cycles with 5 ≤ a = b ≤ 8.

Runrun Liu DP-coloring of planar graphs

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Results 2

Some results on list 3-coloring can be strengthened: Theorem (L.-Loeb-Yin-Yu, 2019) Planar graphs without {3, 6, 7, 8}-cycles, or {3, 5, 6}-cycles or {5, 6, 7}-cycles, or {4, 5, 6, 9}-cycles, or {4, 5, 7, 9}-cycles are DP-3-colorable. In particular, planar graphs without {4, a, b, 9}-cycles with 5 ≤ a = b ≤ 8. Theorem (L.-Loeb-Rolek-Yin-Yu, 2019) Planar graphs without {4, 9}-cycles and cycles of two lengths from {6, 7, 8} are DP-3-colorable.

Runrun Liu DP-coloring of planar graphs

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Results 2

Some results on list 3-coloring can be strengthened: Theorem (L.-Loeb-Yin-Yu, 2019) Planar graphs without {3, 6, 7, 8}-cycles, or {3, 5, 6}-cycles or {5, 6, 7}-cycles, or {4, 5, 6, 9}-cycles, or {4, 5, 7, 9}-cycles are DP-3-colorable. In particular, planar graphs without {4, a, b, 9}-cycles with 5 ≤ a = b ≤ 8. Theorem (L.-Loeb-Rolek-Yin-Yu, 2019) Planar graphs without {4, 9}-cycles and cycles of two lengths from {6, 7, 8} are DP-3-colorable. We cannot (yet) strengthen the case without {4, 5, 8, 9}-cycle.

Runrun Liu DP-coloring of planar graphs

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Results 3

By way of vertex identification, new sufficient conditions for a graph to be 4-choosable (and DP-4-coloring) can be obtained

Runrun Liu DP-coloring of planar graphs

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Results 3

By way of vertex identification, new sufficient conditions for a graph to be 4-choosable (and DP-4-coloring) can be obtained Theorem (L.-Li, 2019) Planar graphs without 4-cycles adjacent to two triangles are DP-4-colorable.

Runrun Liu DP-coloring of planar graphs

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Results 3

By way of vertex identification, new sufficient conditions for a graph to be 4-choosable (and DP-4-coloring) can be obtained Theorem (L.-Li, 2019) Planar graphs without 4-cycles adjacent to two triangles are DP-4-colorable. Theorem (Chen-L.-Yu-Zhao-Zhou, 2018+) Planar graphs without 4-cycles adjacent to 5-cycles, or 4-cycles adjacent to 6-cycles are DP-4-colorable.

Runrun Liu DP-coloring of planar graphs

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Results 3

By way of vertex identification, new sufficient conditions for a graph to be 4-choosable (and DP-4-coloring) can be obtained Theorem (L.-Li, 2019) Planar graphs without 4-cycles adjacent to two triangles are DP-4-colorable. Theorem (Chen-L.-Yu-Zhao-Zhou, 2018+) Planar graphs without 4-cycles adjacent to 5-cycles, or 4-cycles adjacent to 6-cycles are DP-4-colorable. Theorem (L.-Li-Nakprasit-Sittitrai-Yu, 2018+) Planar graphs without 3-cycles adjacent to 5-cycles, or 3-cycles adjacent to 6-cycles are DP-4-colorable.

Runrun Liu DP-coloring of planar graphs

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Future directions

Runrun Liu DP-coloring of planar graphs

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Future directions

Is planar graph without cycles of length from 4 to 8 DP-3-colorable?

Runrun Liu DP-coloring of planar graphs

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Future directions

Is planar graph without cycles of length from 4 to 8 DP-3-colorable? Is planar graph without K −

4 4-choosable? DP-4-colorable?

Runrun Liu DP-coloring of planar graphs

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Future directions

Is planar graph without cycles of length from 4 to 8 DP-3-colorable? Is planar graph without K −

4 4-choosable? DP-4-colorable?

Is every simple graph DP-edge-colorable with ∆ + 1 colors?

Runrun Liu DP-coloring of planar graphs

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Future directions

Is planar graph without cycles of length from 4 to 8 DP-3-colorable? Is planar graph without K −

4 4-choosable? DP-4-colorable?

Is every simple graph DP-edge-colorable with ∆ + 1 colors? DP-coloring of hypergraphs (Bernshteyn and Kostochka)

Runrun Liu DP-coloring of planar graphs

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Future directions

Is planar graph without cycles of length from 4 to 8 DP-3-colorable? Is planar graph without K −

4 4-choosable? DP-4-colorable?

Is every simple graph DP-edge-colorable with ∆ + 1 colors? DP-coloring of hypergraphs (Bernshteyn and Kostochka) DP-coloring of digraphs (Bang-Jensen, Bellitto, Schweser, Stiebitz)

Runrun Liu DP-coloring of planar graphs