Lucas Pastor November 10 2017 Joint-work with Rmi de Joannis de - - PowerPoint PPT Presentation

Coloring squares of claw-free graphs

Lucas Pastor

November 10 2017 Joint-work with Rémi de Joannis de Verclos and Ross J. Kang

1

A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

2

A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

2

A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

1 1 2 3

2

A (proper) k-coloring of G is an assignement of colors {1, . . . , k} to the vertices of G such that any two adjacent vertices receive a different color.

1 1 2 3

The chromatic number, χ(G), is the smallest k such that G is k-colorable.

2

A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

3

A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

3

A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

1 2 2 3

3

A (proper) k-edge-coloring of G is an assignment of colors {1, . . . , k} to the edges of G such that any two adjacent edges (sharing a vertex) receive a different color.

1 2 2 3

The chromatic index, χ′(G), is the smallest k such that G is k-edge-colorable.

3

Note that in an edge coloring, each color class is a matching.

4

Note that in an edge coloring, each color class is a matching.

1

4

Note that in an edge coloring, each color class is a matching.

3

4

Note that in an edge coloring, each color class is a matching.

2 2

4

Note that in an edge coloring, each color class is a matching.

2 2

But not necessarily an induced matching!

4

A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

5

A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

5

A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

1 2 3 4

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A strong k-edge-coloring of G is a k-edge-coloring where each color class is an induced matching.

1 2 3 4

The strong chromatic index, χ′

s(G), is the smallest k such that

G is strong k-edge-colorable.

5

Questions Given a graph G with maximum degree ∆(G).

6

Questions Given a graph G with maximum degree ∆(G). χ′

s(G) 6

Questions Given a graph G with maximum degree ∆(G). χ′

s(G) ≤ upper bound 6

Questions Given a graph G with maximum degree ∆(G). lower bound ≤ χ′

s(G) ≤ upper bound 6

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

7

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e

7

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e

7

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

7

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

∆ − 1

7

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

∆ − 1

∆ ∆

∆ − 1

7

Upper bound Pick any edge e, and look at how large can be its neighborhood at distance 2.

e ∆ ∆

∆ − 1

∆ ∆

∆ − 1

χ′

s(G) ≤ 2∆(∆ − 1) + 1 = 2∆2 − 2∆ + 1. 7

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

In this graph, any pair of edges is at distance at most 2. There are

5 4∆2 edges in G. 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

1 4 ∆2

In this graph, any pair of edges is at distance at most 2. There are

5 4∆2 edges in G. 8

Lower bound For any even integer ∆ ≥ 2, there exist a graph G of max degree ∆ such that: χ′

s(G) = 5

4∆2.

∆ 2 ∆ 2 ∆ 2 ∆ 2 ∆ 2

1 4 ∆2 1 4 ∆2 1 4 ∆2 1 4 ∆2 1 4 ∆2

In this graph, any pair of edges is at distance at most 2. There are

5 4∆2 edges in G. 8

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2. 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better?

9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better? Theorem [Molloy, Reed 1997] χ′

s(G) ≤ (2 − ǫ)∆(G)2 9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better? Theorem [Molloy, Reed 1997] χ′

s(G) ≤ (2 − ǫ)∆(G)2

for some constant ǫ = 0.002.

9

Conjecture [Erdős, Nešetřil 1988] The previous example is the worst you can get. In other words: For any graph G, χ′

s(G) ≤ 5 4∆(G)2.

We have an upper bound of 2∆(G)2. Can we do better? Theorem [Molloy, Reed 1997] χ′

s(G) ≤ (2 − ǫ)∆(G)2

for some constant ǫ = 0.002. The constant has been improved by Bruhn and Joos in 2015 to ǫ = 0.07.

9

Line-graph Given a graph G, the line-graph of G, denoted by L(G), is the graph whose vertices are the edges of G and whose edges are the pairs of adjacent edges of G.

10

Line-graph Given a graph G, the line-graph of G, denoted by L(G), is the graph whose vertices are the edges of G and whose edges are the pairs of adjacent edges of G.

e1 e2 e3 e4 e5 e6 G e1 e2 e4 e5 e6 e3 L(G)

10

Line-graph Given a graph G, the line-graph of G, denoted by L(G), is the graph whose vertices are the edges of G and whose edges are the pairs of adjacent edges of G.

e1 e2 e3 e4 e5 e6 G e1 e2 e4 e5 e6 e3 L(G)

Note than if G is a simple graph, then ω(L(G)) = ∆(G) unless G is the disjoint union of a triangle, paths and cyles.

10

Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance at most 2.

11

Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance at most 2.

G

11

Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance at most 2.

G G2

11

Square graph Given a graph G, the square of G, denoted by G2, is the graph

• btained from G by adding edges between every pair of vertices at

distance at most 2.

G G2

11

Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).

12

Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).
• The strong coloring of G is equivalent to color G2.

12

Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).
• The strong coloring of G is equivalent to color G2.
• Hence, the strong edge coloring of G is equivalent to color the

vertices of L(G)2.

12

Strong coloring

• Coloring the edges of G is equivalent to coloring the vertices
• f L(G).
• The strong coloring of G is equivalent to color G2.
• Hence, the strong edge coloring of G is equivalent to color the

vertices of L(G)2. Molloy and Reed’s theorem Let G be the line-graph of any simple graph, then: χ(G2) ≤ (2 − ǫ)ω(G)2.

12

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

13

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

13

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

13

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques.

13

Line-graphs In a line-graph, the neighborhood of any vertex is the union of at most 2 cliques. The class of graphs having this property is the class of quasi-line graphs.

13

Quasi-line graphs In a quasi-line graph, the neighborhood of any vertex cannot have 3 pairwise non-adjacent vertices.

14

Quasi-line graphs In a quasi-line graph, the neighborhood of any vertex cannot have 3 pairwise non-adjacent vertices.

claw

14

Quasi-line graphs In a quasi-line graph, the neighborhood of any vertex cannot have 3 pairwise non-adjacent vertices.

claw

The class of graphs having this property is the class of claw-free graphs.

14

line-graph

15

line-graph quasi-line

15

line-graph quasi-line claw-free

15

line-graph quasi-line claw-free Molloy and Reed Molloy and Reed

15

line-graph quasi-line claw-free Molloy and Reed Molloy and Reed Us

15

Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2

16

Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2 Roadmap

• 1. From claw-free to quasi-line graphs.

16

Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2 Roadmap

• 1. From claw-free to quasi-line graphs.
• 2. From quasi-line graphs to line-graphs of multigraphs.

16

Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G: χ(G2) ≤ (2 − ǫ)ω(G)2 Roadmap

• 1. From claw-free to quasi-line graphs.
• 2. From quasi-line graphs to line-graphs of multigraphs.
• 3. Prove the theorem for line-graphs of multigraphs.

16

Second neighborhood The second neighborhood of v, denoted by N2

G(v), is the set of

vertices at distance exactly two from v, i.e. N2

G(v) = NG2(v) \ NG(v). 17

Second neighborhood The second neighborhood of v, denoted by N2

G(v), is the set of

vertices at distance exactly two from v, i.e. N2

G(v) = NG2(v) \ NG(v).

The square degree of v, denoted by degG2(v), is equal to degG(v) + |N2

G(v)|. 17

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

18

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

• 1. The proof is by induction on |V (G)|.

18

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

• 1. The proof is by induction on |V (G)|.
• 2. Note that (G \ v)2 = G2 \ v.

18

v

19

v

19

v

19

19

19

v

19

v

19

v clique in G2 − v

19

maybe not in (G \ v)2

19

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2.

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) ≥ 3

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) ≥ 3

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) ≥ 3

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V (G) with degG2(v) ≤ ω(G)2 + (ω(G) + 1)/2 whose neighborhood is a clique of (G \ v)2. If NG(v) is not a clique of (G \ v)2 then NG(v) is the union of two cliques.

v x y d(x, y) = 3 z

20

Upper bound on degG2(v).

21

Upper bound on degG2(v).

v

21

Upper bound on degG2(v).

v Let u with minimum |N(u) ∩ N(v)| = k u

21

Upper bound on degG2(v).

v Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v) w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w C2 = N(v) \ (N(u) ∪ N(w)) Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1 C2

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w C2 = N(v) \ (N(u) ∪ N(w)) Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1 C2

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w C2 = N(v) \ (N(u) ∪ N(w)) Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1 C2

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w C2 = N(v) \ (N(u) ∪ N(w)) Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1 C2

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w C2 = N(v) \ (N(u) ∪ N(w)) Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1 C2

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w C2 = N(v) \ (N(u) ∪ N(w)) Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1 C2

w

21

Upper bound on degG2(v).

v X = N(v) ∩ N(u) \ w C1 = (N(v) ∩ N(w) \ X) ∪ w C2 = N(v) \ (N(u) ∪ N(w)) Let u with minimum |N(u) ∩ N(v)| = k u Let w ∈ N(u) ∩ N(v)

X C1 C2

w

21

Upper bound on degG2(v).

v degG2(v) ≤ degG(v) + |N 2(v)|

X C1 C2

w u

21

Upper bound on degG2(v).

v Count #P3 from v to |N 2(v)| degG2(v) ≤ degG(v) + |N 2(v)|

X C1 C2

w u

21

Upper bound on degG2(v).

v |N 2(v)| ≤ #P3 ≤ degG(v)(ω − 1) Count #P3 from v to |N 2(v)| degG2(v) ≤ degG(v) + |N 2(v)|

X C1 C2

w u

21

Upper bound on degG2(v).

v |N 2(v)| ≤ #P3 ≤ degG(v)(ω − 1) Count #P3 from v to |N 2(v)| degG2(v) ≤ degG(v) + |N 2(v)|

X C1 C2

w u Every vertex of N 2(v) has degree at least k #P3 ≥ k|N 2(v)|

21

Upper bound on degG2(v).

v |N 2(v)| ≤ #P3 ≤ degG(v)(ω − 1)/k Count #P3 from v to |N 2(v)|

X C1 C2

w u Every vertex of N 2(v) has degree at least k degG2(v) ≤ degG(v) + |N 2(v)| #P3 ≥ k|N 2(v)|

21

Upper bound on degG2(v).

v Count #P3 from v to |N 2(v)| degG2(v) ≤ (1 + ω−1

k )degG(v)

Every vertex of N 2(v) has degree at least k

X C1 C2

w u |N 2(v)| ≤ #P3 ≤ degG(v)(ω − 1)/k #P3 ≥ k|N 2(v)|

21

Upper bound on degG2(v).

if k = 1, N(v) is the union of two cliques v degG2(v) ≤ (1 + ω−1

k )degG(v)

X C1 C2

w u degG(v) ≤ 2ω + k

21

Upper bound on degG2(v).

if k = 1, N(v) is the union of two cliques if k ≥ 2, degG2(v) ≤ ω2 + O(ω) v degG2(v) ≤ (1 + ω−1

k )degG(v)

X C1 C2

w u degG(v) ≤ 2ω + k

21

Lemma From quasi-line graphs to line-graphs of multigraphs.

22

Lemma From quasi-line graphs to line-graphs of multigraphs.

• Structure theorem of claw-free graphs due to Chudnovsky and

Seymour.

22

Lemma From quasi-line graphs to line-graphs of multigraphs.

• Structure theorem of claw-free graphs due to Chudnovsky and

Seymour.

• Either there is a good vertex, or G is the line-graph of a

multigraph.

22

Lemma For G line-graph of multigraph, there is an absolute constant ǫ > 0 such that χ(G2) ≤ (2 − ǫ)ω(G)2.

23

Lemma For G line-graph of multigraph, there is an absolute constant ǫ > 0 such that χ(G2) ≤ (2 − ǫ)ω(G)2. The idea is to generalize the proof of Molloy and Reed to line graphs of multigraphs.

23

Molloy and Reed For any ǫ > 0, there exist δ > 0 and ∆0 such that the following

• holds. For all ∆ ≥ ∆0, if G is a graph with ∆(G) ≤ ∆ and with at

most (1 − ǫ)

∆

2

edges in each neighbourhood, then

χ(G) ≤ (1 − δ)∆.

24

Molloy and Reed For any ǫ > 0, there exist δ > 0 and ∆0 such that the following

• holds. For all ∆ ≥ ∆0, if G is a graph with ∆(G) ≤ ∆ and with at

most (1 − ǫ)

∆

2

edges in each neighbourhood, then

χ(G) ≤ (1 − δ)∆. If the neighborhood is not too dense, then the chromatic number is not too big.

24

Theorem There are some absolute constants ǫ > 0 and ∆0 such that χ′

s(F) ≤ (2 − ǫ)∆(F)2 for any multigraph F with ∆(F) ≥ ∆0. 25

Theorem There are some absolute constants ǫ > 0 and ∆0 such that χ′

s(F) ≤ (2 − ǫ)∆(F)2 for any multigraph F with ∆(F) ≥ ∆0.

Lemma There are absolute constants ǫ > 0 and ∆0 such that the following

• holds. For all ∆ ≥ ∆0, if F = (V , E) is a multigraph with

∆(F) ≤ ∆, then NL(F)2(e) induces a subgraph of L(F)2 with at most (1 − ǫ)

2∆(∆−1)

2

edges for any e ∈ E.

25

Theorem There are some absolute constants ǫ > 0 and ∆0 such that χ′

s(F) ≤ (2 − ǫ)∆(F)2 for any multigraph F with ∆(F) ≥ ∆0.

Lemma There are absolute constants ǫ > 0 and ∆0 such that the following

• holds. For all ∆ ≥ ∆0, if F = (V , E) is a multigraph with

∆(F) ≤ ∆, then NL(F)2(e) induces a subgraph of L(F)2 with at most (1 − ǫ)

2∆(∆−1)

2

edges for any e ∈ E.

Since ∆(L(F)2) ≤ 2∆(F)(∆(F) − 1), we apply the theorem of Molloy and Reed to L(F)2.

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How to bound the edge density of NL(F)2(e)?

26

How to bound the edge density of NL(F)2(e)?

u1 u2 e

26

How to bound the edge density of NL(F)2(e)?

u1 u2 e A B

26

How to bound the edge density of NL(F)2(e)?

u1 u2 e A B C

26

How to bound the edge density of NL(F)2(e)?

u1 u2 e A B C

26

How to bound the edge density of NL(F)2(e)?

u1 u2 e A B C

26

How to bound the edge density of NL(F)2(e)?

u1 u2 e A B C

26

How to bound the edge density of NL(F)2(e)?

u1 u2 e A B C

26

Conclusion

• Our constant can be improved by using Bruhn and Joos

method.

27

Conclusion

• Our constant can be improved by using Bruhn and Joos

method.

• The conjecture for bipartite graphs is χ′

s(G) ≤ ∆(A)∆(B). 27

Thank you for your attention.

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