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

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

Molloy and Reed Molloy and Reed line-graph quasi-line claw-free 15

Molloy and Reed Molloy and Reed line-graph quasi-line Us claw-free 15

Theorem [de Joannis de Verclos, Kang, P.] There is an absolute constant ǫ > 0 such that, for any claw-free graph G : χ ( G 2 ) ≤ (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 ( G 2 ) ≤ (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 ( G 2 ) ≤ (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 ( G 2 ) ≤ (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 N 2 G ( v ), is the set of vertices at distance exactly two from v , i.e. N 2 G ( v ) = N G 2 ( v ) \ N G ( v ). 17

Second neighborhood The second neighborhood of v , denoted by N 2 G ( v ), is the set of vertices at distance exactly two from v , i.e. N 2 G ( v ) = N G 2 ( v ) \ N G ( v ). The square degree of v , denoted by deg G 2 ( v ), is equal to deg G ( v ) + | N 2 G ( v ) | . 17

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( 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 deg G 2 ( 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 deg G 2 ( 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 � = G 2 \ v . 18

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clique in G 2 − v 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 deg G 2 ( 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 deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( 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 deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( 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 deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. y x v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) ≥ 3 y x v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) ≥ 3 y x v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) ≥ 3 y x v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x z v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x z v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x z v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x z v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x z v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x z v 20

Lemma For G claw-free, either G is a quasi-line graph or there exist v ∈ V ( G ) with deg G 2 ( v ) ≤ ω ( G ) 2 + ( ω ( G ) + 1) / 2 whose neighborhood is a clique of ( G \ v ) 2 . If N G ( v ) is not a clique of ( G \ v ) 2 then N G ( v ) is the union of two cliques. d ( x, y ) = 3 y x z v 20

Upper bound on deg G 2 ( v ). 21

Upper bound on deg G 2 ( v ). v 21