Locally identifying colorings of graphs Aline Parreau Joint work - - PowerPoint PPT Presentation

Locally identifying colorings of graphs

Aline Parreau

Joint work with Louis Esperet, Sylvain Gravier, Micka¨ el Montassier, Pascal Ochem Turku, March 28th, 2011 ANR IDEA

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Grenoble-Turku

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Outline

Proper Colorings Definition Perfect graphs Colorings of hypergraph Locally Identifying Colorings Definition Bipartite graphs Perfect graphs are not any more perfect Planar graphs

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Proper coloring

Graph G = (V , E) Proper coloring : c : V → N s.t. uv ∈ E ⇒ c(u) = c(v)

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Proper coloring

Graph G = (V , E) Proper coloring : c : V → N s.t. uv ∈ E ⇒ c(u) = c(v) 1 2 3 4 5 6 1 2 3 4 5 1

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A well-known example

4-Colour Theorem : But also : frequency allocation, scheduling,...

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A well-known example

4-Colour Theorem : But also : frequency allocation, scheduling,...

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A well-known example

4-Colour Theorem : But also : frequency allocation, scheduling,...

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Chromatic number

Chromatic number χ(G) : minimum number of colors to have a porper coloring : 1 2 3 4 5 6 1 2 3 4 5 1 χ(G) ≤ 6

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Chromatic number

Chromatic number χ(G) : minimum number of colors to have a porper coloring : 1 2 3 4 5 6 1 2 3 4 5 1 1 2 3 4 5 5 1 2 3 4 5 1 χ(G) ≤ 5

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Chromatic number

Chromatic number χ(G) : minimum number of colors to have a porper coloring : 1 2 3 4 5 6 1 2 3 4 5 1 1 2 3 2 3 2 3 3 1 2 3 1 χ(G) = 3

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A lower bound...

Clique number ω(G) : max k such that there are k vertices in G that are all connected to each other ω(G) = 3

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A lower bound...

Clique number ω(G) : max k such that there are k vertices in G that are all connected to each other ω(G) = 4

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A lower bound...

Clique number ω(G) : max k such that there are k vertices in G that are all connected to each other ω(G) = 4

For any graph G, χ(G) ≥ ω(G)

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... that is not always reached

χ(C5) = 3 but ω(C5) = 2

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... that is not always reached

1 2 1 2 3 χ(C5) = 3 but ω(C5) = 2

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... that is not always reached

1 2 1 2 3 χ(C5) = 3 but ω(C5) = 2 Mycielski graphs Mk such that χ(Mk) = k but ω(Mk) = 2

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An upper bound

∆(G) : maximum degree of G Upper bound with greedy algorithm : χ(G) ≤ ∆(G) + 1 Tight for : complete graphs, odd cycles Brook’s theorem (1941) : χ(G) ≤ ∆(G) if G is not a complete graph or an odd cycle

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It’s hard to guess !

k-Coloring is NP-complete for any k ≥ 3 Even for k = 3 and planar graphs with maximum degree 4 !

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Good graphs for coloring

Perfect graph (1963) : G is perfect if ω(H) = χ(H) for any induced subgraph H of G Examples :

• trees (and bipartite graphs) :
• interval graphs :

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Strong Perfect Graph Theorem

Remark : perfect graphs are stable by induced subgraphs Smallest non perfect graphs ? → odd cycles of size ≥ 5, complement of odd cycles of size ≥ 5 Strong Perfect Graph Theorem (Chudnovsky, Robertson, Seymour, Thomas 2002) : G is perfect if and only if it has no induced odd cycle or complement of odd cycle with more than 5 vertices

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A part of the big family of perfect graphs Perfect

Chordal Permutation Line of bipartite Cograph Trees k-trees Split Bipartite Interval

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k-trees

Construction of 2-trees :

• Start with a triangle

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k-trees

Construction of 2-trees :

• Start with a triangle
• Choose an edge

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k-trees

Construction of 2-trees :

• Start with a triangle
• Choose an edge
• Add a vertex connected to

the 2 vertices of the edge

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k-trees

Construction of 2-trees :

• Start with a triangle
• Choose an edge
• Add a vertex connected to

the 2 vertices of the edge

• Repeat the operation

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k-trees

Construction of 2-trees :

• Start with a triangle
• Choose an edge
• Add a vertex connected to

the 2 vertices of the edge

• Repeat the operation

Remark : any edge is in a triangle

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k-trees

Construction of 2-trees :

• Start with a triangle
• Choose an edge
• Add a vertex connected to

the 2 vertices of the edge

• Repeat the operation

Remark : any edge is in a triangle

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k-trees

Construction of 2-trees :

• Start with a triangle
• Choose an edge
• Add a vertex connected to

the 2 vertices of the edge

• Repeat the operation

Remark : any edge is in a triangle

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k-trees

Construction of 2-trees :

• Start with a triangle
• Choose an edge
• Add a vertex connected to

the 2 vertices of the edge

• Repeat the operation

Remark : any edge is in a triangle

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Hypergraph

Hypergraph H = (V , E) where an edge e ∈ E is a subset of vertices

E D C B A e1 = {A, E} e2 = {A, B} e3 = {B, C, D} e4 = {C, D, E}

Hypergraph

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Hypergraph

Hypergraph H = (V , E) where an edge e ∈ E is a subset of vertices

E D C B A e1 = {A, E} e2 = {A, B} e3 = {B, C, D} e4 = {C, D, E}

Hypergraph E D C B A

e4 e3 e2 e1

Corresponding bipartite graph

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Colorings of Hypergraph

c : V → N is a proper coloring of H if no edge is unicolor.

E D C B A

A 2-proper coloring 2-Hypergraph-Coloring is NP-complete

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Locally identifying coloring (lid-coloring)

In proper coloring : Two adjacent vertices have distinct colors . 1 1 2 1 2 2

Bt(u) = {v | d(u, v) ≤ t}

For any edge uv, c(B0(u)) = c(B0(v))

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Locally identifying coloring (lid-coloring)

In proper coloring : Two adjacent vertices have distinct colors . 1 1 2 1 2 2

Bt(u) = {v | d(u, v) ≤ t}

For any edge uv, c(B0(u)) = c(B0(v))

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Locally identifying coloring (lid-coloring)

In locally identifying coloring : Two adjacent vertices have distinct colors in their neighborhood. 1 1 2 1 2 2

Bt(u) = {v | d(u, v) ≤ t}

For any edge uv, c(B0(u)) = c(B0(v)) and c(B1(u)) = c(B1(v))

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Locally identifying coloring (lid-coloring)

In locally identifying coloring : Two adjacent vertices have distinct colors in their neighborhood. 1 1 2 3 4 4

1, 2, 3 2, 3, 4

Bt(u) = {v | d(u, v) ≤ t}

For any edge uv, c(B0(u)) = c(B0(v)) and c(B1(u)) = c(B1(v))

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Locally identifying coloring (lid-coloring)

In locally identifying coloring : Two adjacent vertices have distinct colors in their neighborhood. 1 1 2 3 4 4

1, 2, 3 2, 3, 4

Bt(u) = {v | d(u, v) ≤ t}

For any edge uv, c(B0(u)) = c(B0(v)) and c(B1(u)) = c(B1(v)) whenever B1(u) = B1(v)

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Locally identifying coloring (lid-coloring)

In locally identifying coloring : Two adjacent vertices have distinct colors in their neighborhood. 1 1 2 3 4 4

1, 2, 3 2, 3, 4

Bt(u) = {v | d(u, v) ≤ t}

For any edge uv, c(B0(u)) = c(B0(v)) and c(B1(u)) = c(B1(v)) whenever B1(u) = B1(v) χlid(G) : lid-chromatic number

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An example : the path

With 4 colors :

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4 Is it possible with 3 colors ?

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4 Is it possible with 3 colors ?

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4 Is it possible with 3 colors ? 1 2

1, 2

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4 Is it possible with 3 colors ? 1 2

1, 2

3

1, 2, 3

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4 Is it possible with 3 colors ? 1 2

1, 2

3

1, 2, 3

2

2, 3

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4 Is it possible with 3 colors ? 1 2

1, 2

3

1, 2, 3

2

2, 3

1

1, 2, 3

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An example : the path

With 4 colors : 1 2 3 4 1 2 3 4

1, 2 1, 2, 3 2, 3, 4 1, 3, 4 1, 2, 4 1, 2, 3 2, 3, 4 3, 4

So : χlid(Pk) ≤ 4 Is it possible with 3 colors ? 1 2

1, 2

3

1, 2, 3

2

2, 3

1

1, 2, 3

2 3 2

1, 2 1, 2, 3 2, 3 2, 3

χlid(Pk) = 3 ⇔ k is odd

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Related works

With edge colorings :

• Vertex-distinguishing edge colorings (Observability of a graph)

(Hornak et al, 95’),

• Adjacent vertex-distinguishing edge colorings (Zhang et al,

02’) With total colorings :

• Adjacent vertex-distinguishing total colorings (Zhang, 05’)

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Link with chromatic number

χlid(G) ≥ χ(G) Do we need much more than χ(G) colors ?

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Link with chromatic number

χlid(G) ≥ χ(G) Do we need much more than χ(G) colors ?

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Link with chromatic number

χlid(G) ≥ χ(G) Do we need much more than χ(G) colors ? An example with χ(G) = 3 and χlid(G) ≥ k

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Link with chromatic number

χlid(G) ≥ χ(G) Do we need much more than χ(G) colors ? 1 1 An example with χ(G) = 3 and χlid(G) ≥ k

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Link with chromatic number

χlid(G) ≥ χ(G) Do we need much more than χ(G) colors ? 1 1 2 3 An example with χ(G) = 3 and χlid(G) ≥ k

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Link with chromatic number

χlid(G) ≥ χ(G) Do we need much more than χ(G) colors ? 1 1 2 3 1, 2, 3 ← 1, 2, 3 ← An example with χ(G) = 3 and χlid(G) ≥ k

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Link with chromatic number

χlid(G) ≥ χ(G) Do we need much more than χ(G) colors ? An example with χ(G) = 3 and χlid(G) ≥ k χlid is not bounded by a function of χ But...

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Link with maximum degree

We have : χlid(G) ≤ χ(G 3) This implies : χlid(G) ≤ ∆(G)3 − ∆(G)2 + ∆(G) + 1

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Link with maximum degree

We have : χlid(G) ≤ χ(G 3) This implies : χlid(G) ≤ ∆(G)3 − ∆(G)2 + ∆(G) + 1 We know only graph that needs ∆(G)2 + ∆(G) + 1

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Transition Slide

No bounds with χ for general graphs..

What about “good classes” for proper colorings ?

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Perfect graphs Perfect

Chordal Permutation Line of bipartite Cograph Trees k-trees Split Bipartite Interval Trees Bipartite ? ?

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An amazing fact about bipartite graphs

G connected graph :

• χlid(G) = 1 ⇒ G is a single vertex

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An amazing fact about bipartite graphs

G connected graph :

• χlid(G) = 1 ⇒ G is a single vertex
• χlid(G) = 2 ⇒ G is just an edge

1 1, 2 2 1, 2

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An amazing fact about bipartite graphs

G connected graph :

• χlid(G) = 1 ⇒ G is a single vertex
• χlid(G) = 2 ⇒ G is just an edge

1 1, 2 3 2 1, 2

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An amazing fact about bipartite graphs

G connected graph :

• χlid(G) = 1 ⇒ G is a single vertex
• χlid(G) = 2 ⇒ G is just an edge

1 1, 2 3 2 1, 2

• χlid(G) = 3 ⇒ G is a triangle or a bipartite graph :

→ Partition vertices with the number of colors they see

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Bipartite graphs

L0 L1 L2 L3 L4

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Bipartite graphs

L0 L1 L2 L3 L4 1 2 3 4 1 → → → → →

1, 2 1, 2, 3 2, 3, 4 or 2, 3 1, 3, 4 or 3, 4 1, 4 25/40

Bipartite graphs

General bounds : 3 ≤ χlid(B) ≤ 4

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Bipartite graphs

General bounds : 3 ≤ χlid(B) ≤ 4

χlid(B) = 3 :

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Bipartite graphs

General bounds : 3 ≤ χlid(B) ≤ 4

χlid(B) = 3 : χlid(B) = 4 :

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Bipartite graphs

General bounds : 3 ≤ χlid(B) ≤ 4

χlid(B) = 3 : χlid(B) = 4 :

← ? → In general... 3-Lid-Coloring is NP-complete in bipartite graphs

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Link with 2-coloring of hypergraph

Try to color a graph with 3 colors

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Link with 2-coloring of hypergraph

Try to color a graph with 3 colors

1, 2, 3

1 2 3

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Link with 2-coloring of hypergraph

Try to color a graph with 3 colors

1, 2, 3 1, 2 1, 3

1 1 1 2 3

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Link with 2-coloring of hypergraph

Try to color a graph with 3 colors

1, 2, 3 1, 2 1, 3 1, 2 1, 2 1, 3 1, 2, 3 1, 2, 3 1, 2, 3

2 2 3 1 1 1 1 2 3

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Link with 2-coloring of hypergraph

Try to color a graph with 3 colors

1, 2, 3 1, 2 1, 3 1, 2 1, 2 1, 3 1, 2, 3 1, 2, 3 1, 2, 3

E D C B A

E D C B A

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Link with 2-coloring of hypergraph

Try to color a graph with 3 colors

1, 2, 3 1, 2 1, 3 1, 2 1, 2 1, 3 1, 2, 3 1, 2, 3 1, 2, 3

E D C B A

E D C B A

• 3-Lid-Coloring in bipartite graph is NP-Complete
• Polynomial if B regular, if B is planar with maximum degree

3, if B is a tree.

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Perfect graphs Perfect

Chordal Permutation Line of bipartite Cograph Trees k-trees Split Bipartite Interval Trees Bipartite k-trees

≤ 4 = 2ω ≤ 4 = 2ω

?

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To perfect graph : k-trees

Lid-coloring of 2-trees with 6 colors :

1 2 3

• Color the triangle with

colors 1, 2, 3

• Step :

i

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To perfect graph : k-trees

Lid-coloring of 2-trees with 6 colors :

1 2 3 5

• Color the triangle with

colors 1, 2, 3

• Step :

i i + 3[6]

• We always have :

◮ proper coloring ◮ no edge (i, i + 3) 29/40

To perfect graph : k-trees

Lid-coloring of 2-trees with 6 colors :

1 2 3 5 6

• Color the triangle with

colors 1, 2, 3

• Step :

i i + 3[6]

• We always have :

◮ proper coloring ◮ no edge (i, i + 3) 29/40

To perfect graph : k-trees

Lid-coloring of 2-trees with 6 colors :

1 2 3 5 6 4

• Color the triangle with

colors 1, 2, 3

• Step :

i i + 3[6]

• We always have :

◮ proper coloring ◮ no edge (i, i + 3) 29/40

To perfect graph : k-trees

Lid-coloring of 2-trees with 6 colors :

1 2 3 5 6 4 4

• Color the triangle with

colors 1, 2, 3

• Step :

i i + 3[6]

• We always have :

◮ proper coloring ◮ no edge (i, i + 3) 29/40

To perfect graph : k-trees

Lid-coloring of 2-trees with 6 colors :

1 2 3 5 6 4 4 6 5 3 4 2 3

• Color the triangle with

colors 1, 2, 3

• Step :

i i + 3[6]

• We always have :

◮ proper coloring ◮ no edge (i, i + 3) 29/40

To perfect graph : k-trees

We can extend the construction to k-trees : → A k-tree has lid-chromatic number at most 2k + 2 This bound is sharp : Pk

2k+2

1 2 3 4 5 6 7 8

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To perfect graph : k-trees

We can extend the construction to k-trees : → A k-tree has lid-chromatic number at most 2k + 2 This bound is sharp : Pk

2k+2

1 2 3 4 5 6 7 8 1 2 3 4 Complete graph

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To perfect graph : k-trees

We can extend the construction to k-trees : → A k-tree has lid-chromatic number at most 2k + 2 This bound is sharp : Pk

2k+2

1 2 3 4 5 6 7 8 1 2 3 4 5 Separated by vertex 5

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To perfect graph : k-trees

We can extend the construction to k-trees : → A k-tree has lid-chromatic number at most 2k + 2 This bound is sharp : Pk

2k+2

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

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Perfect Graphs Perfect

Chordal Permutation Line of bipartite Cograph Trees k-trees Split Bipartite Interval Trees

≤ 4 = 2ω

Bipartite

≤ 4 = 2ω

k-trees

≤ 2k + 2 = 2ω

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Perfect Graphs Perfect

Chordal Permutation Line of bipartite Cograph Trees k-trees Split Bipartite Interval Trees

≤ 4 = 2ω

Bipartite

≤ 4 = 2ω

k-trees

≤ 2k + 2 = 2ω

Split Interval Cograph

≤ 2ω − 1 ≤ 2ω − 1 ≤ 2ω

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Perfect Graphs Perfect

Chordal Permutation Line of bipartite Cograph Trees k-trees Split Bipartite Interval Trees

≤ 4 = 2ω

Bipartite

≤ 4 = 2ω

k-trees

≤ 2k + 2 = 2ω

Split Interval Cograph

≤ 2ω − 1 ≤ 2ω − 1 ≤ 2ω

Perfect

Chordal ?

?

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Perfect graphs are not any more perfect...

Question : Can we color any perfect graph G with 2ω(G) colors ?

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Perfect graphs are not any more perfect...

Question : Can we color any perfect graph G with 2ω(G) colors ? No ! V1 V2 V3

M Kk,k \ M Kk,k

V1, V2, V3 stable sets of size k

χlid ≥ k + 2 but ω = 3

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Perfect graphs are not any more perfect...

Question : Can we color any perfect graph G with 2ω(G) colors ? No ! V1 V2 V3

M Kk,k \ M Kk,k

V1, V2, V3 stable sets of size k

χlid ≥ k + 2 but ω = 3 Conjecture : We can color any chordal graph G with 2ω(G) colors

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Perfect Graphs Perfect

Chordal Permutation Line of bipartite Cograph Trees k-trees Split Bipartite Interval Trees

≤ 2ω

Bipartite ≤ 2ω k-trees

≤ 2ω

Split Interval Cograph

≤ 2ω ≤ 2ω ≤ 2ω

Perfect

Chordal NO ?

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To support the conjecture : Split graphs

Chordal graph : constructed like k-trees but the size of the clique can change

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To support the conjecture : Split graphs

Chordal graph : constructed like k-trees but the size of the clique can change Split graphs :

Kk Independant set

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To support the conjecture : Split graphs

Chordal graph : constructed like k-trees but the size of the clique can change Split graphs :

• Bondy’s theorem : k − 1

vertices of the stable set are enough to separate the clique vertices k − 1

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To support the conjecture : Split graphs

Chordal graph : constructed like k-trees but the size of the clique can change Split graphs :

• Bondy’s theorem : k − 1

vertices of the stable set are enough to separate the clique vertices → We can color with 2k colors → Possible with 2k − 1 colors → It’s sharp

k couleurs

k − 1 1

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Planar graphs

Is lid-chromatic number bounded for planar graphs ?

• Worse example : 8 colors,

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Planar graphs

Is lid-chromatic number bounded for planar graphs ?

• Worse example : 8 colors,
• With large girth (36) bounded by 5

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Planar graphs

Is lid-chromatic number bounded for planar graphs ?

• Worse example : 8 colors,
• With large girth (36) bounded by 5

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Planar graphs

Is lid-chromatic number bounded for planar graphs ?

• Worse example : 8 colors,
• With large girth (36) bounded by 5

Outerplanar graphs :

• General bound : 20 colors,
• Max outerplanar graphs : ≤ 6 colors,
• Without triangles : ≤ 8 colors,
• Examples with at most 6 colors

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A bound for outerplanar graphs

1 2 3 4 1 L1 L2 L3 L4 L5

• a layer = union
• f paths,
• 5 colors in a

layer,

• 4 × 5 = 20

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Bound for planar graphs ?

Really large bound by Gonzcales and Pinlou (2010) More general result : Any family of graph closed by minor has lid-chromatic bounded

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A remark

• For some subclasses of perfect graphs :

χlid(G) ≤ 2ω(G) = 2χ(G)

• For planar graphs, worse example : χlid(G) ≤ 8 = 2χ(G)
• For outerplanar graphs, worse example : χlid(G) ≤ 6 = 2χ(G)
• ...

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A remark

• For some subclasses of perfect graphs :

χlid(G) ≤ 2ω(G) = 2χ(G)

• For planar graphs, worse example : χlid(G) ≤ 8 = 2χ(G)
• For outerplanar graphs, worse example : χlid(G) ≤ 6 = 2χ(G)
• ...

For which graphs do we have χlid(G) ≤ 2χ(G) ? Is it true for planar graphs ?

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Some open problems

• Find a good bound for χlid in planar graphs
• Prove (or disprove) conjecture for chordal graphs
• For which graphs χlid = χ ?
• Better bound with maximum degree ∆ ?
• What about a global version ?

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Kiitos !

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