Disproving the normal graph conjecture Lucas Pastor Joint-work with - - PowerPoint PPT Presentation

The Normal Graph Conjecture Sketch of the proof Conclusion

Disproving the normal graph conjecture Lucas Pastor

Joint-work with Ararat Harutyunyan and Stéphan Thomassé February 08, 2018

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The Normal Graph Conjecture Sketch of the proof Conclusion

Normal graphs Normal graphs appeared in a natural way in the information theory context.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Normal graphs Normal graphs appeared in a natural way in the information theory context. They are also defined in terms of graph theoretical terms.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Normal graphs Normal graphs appeared in a natural way in the information theory context. They are also defined in terms of graph theoretical terms. This class of graphs forms a weaker variant of perfect graphs by means of a specific graph product.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Perfect graph A graph G is perfect if χ(H) = ω(H) for every induced subgraph H of G.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Perfect graph A graph G is perfect if χ(H) = ω(H) for every induced subgraph H of G.

C9, a non perfect graph

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The Normal Graph Conjecture Sketch of the proof Conclusion

Co-normal product The co-normal product G1 ∗ G2 of two graphs G1 and G2 is the graph with vertex set V (G1) × V (G2), where vertices (v1, v2) and (u1, u2) are adjacent if u1 is adjacent to v1 or u2 is adjacent to v2.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Co-normal product The co-normal product G1 ∗ G2 of two graphs G1 and G2 is the graph with vertex set V (G1) × V (G2), where vertices (v1, v2) and (u1, u2) are adjacent if u1 is adjacent to v1 or u2 is adjacent to v2.

u1 v1 u2 v2 G1 G2 w2 (u1, u2) (u1, v2) (u1, w2) (v1, u2) (v1, v2) (v1, w2)

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The Normal Graph Conjecture Sketch of the proof Conclusion

Co-normal product The co-normal product G1 ∗ G2 of two graphs G1 and G2 is the graph with vertex set V (G1) × V (G2), where vertices (v1, v2) and (u1, u2) are adjacent if u1 is adjacent to v1 or u2 is adjacent to v2.

u1 v1 u2 v2 G1 G2 w2 (u1, u2) (u1, v2) (u1, w2) (v1, u2) (v1, v2) (v1, w2)

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The Normal Graph Conjecture Sketch of the proof Conclusion

Berge introduced perfect graphs in 1960. His motivation came in part from the study of the zero-error capacity of a discrete memoryless channel.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Berge introduced perfect graphs in 1960. His motivation came in part from the study of the zero-error capacity of a discrete memoryless channel. Shannon capacity Shannon capacity C(G): C(G) = lim

n→∞

1 n log ω(Gn). Where Gn is the nth co-normal power of G.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Shannon noticed that ω(Gn) = (ω(G))n whenever ω(G) = χ(G).

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The Normal Graph Conjecture Sketch of the proof Conclusion

Shannon noticed that ω(Gn) = (ω(G))n whenever ω(G) = χ(G). One might expect that perfect graphs are closed under co-normal products.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Shannon noticed that ω(Gn) = (ω(G))n whenever ω(G) = χ(G). One might expect that perfect graphs are closed under co-normal products. Körner and Longo 1973 Perfect graphs are not closed under co-normal products.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Definition A graph G is normal if there exists a covering of V (G), C, of cliques and a covering of V (G), S, of stable sets such that C ∩ S = ∅ for every C ∈ C and S ∈ S.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Definition A graph G is normal if there exists a covering of V (G), C, of cliques and a covering of V (G), S, of stable sets such that C ∩ S = ∅ for every C ∈ C and S ∈ S.

clique stable

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The Normal Graph Conjecture Sketch of the proof Conclusion

Definition A graph G is normal if there exists a covering of V (G), C, of cliques and a covering of V (G), S, of stable sets such that C ∩ S = ∅ for every C ∈ C and S ∈ S.

clique stable

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The Normal Graph Conjecture Sketch of the proof Conclusion

Definition A graph G is normal if there exists a covering of V (G), C, of cliques and a covering of V (G), S, of stable sets such that C ∩ S = ∅ for every C ∈ C and S ∈ S.

clique stable

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The Normal Graph Conjecture Sketch of the proof Conclusion

Definition A graph G is normal if there exists a covering of V (G), C, of cliques and a covering of V (G), S, of stable sets such that C ∩ S = ∅ for every C ∈ C and S ∈ S.

clique stable

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The Normal Graph Conjecture Sketch of the proof Conclusion

Definition A graph G is normal if there exists a covering of V (G), C, of cliques and a covering of V (G), S, of stable sets such that C ∩ S = ∅ for every C ∈ C and S ∈ S.

clique stable

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The Normal Graph Conjecture Sketch of the proof Conclusion

Let P be a probability distribution on V (G). We denote by H(G, P) the entropy of G on P, and by H(P) the entropy of P.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Let P be a probability distribution on V (G). We denote by H(G, P) the entropy of G on P, and by H(P) the entropy of P. The graph entropy is sub-additive with respect to complementary graphs: H(P) ≤ H(G, P) + H(G, P).

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The Normal Graph Conjecture Sketch of the proof Conclusion

Let P be a probability distribution on V (G). We denote by H(G, P) the entropy of G on P, and by H(P) the entropy of P. The graph entropy is sub-additive with respect to complementary graphs: H(P) ≤ H(G, P) + H(G, P). Theorem [Csiszár et. al 1990] H(P) = H(G, P) + H(G, P) for all P ⇐ ⇒ G is perfect.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Let P be a probability distribution on V (G). We denote by H(G, P) the entropy of G on P, and by H(P) the entropy of P. The graph entropy is sub-additive with respect to complementary graphs: H(P) ≤ H(G, P) + H(G, P). Theorem [Csiszár et. al 1990] H(P) = H(G, P) + H(G, P) for all P ⇐ ⇒ G is perfect. Theorem [Körner and Marton 1988] H(P) = H(G, P)+H(G, P) for at least one P ⇐ ⇒ G is normal.

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal.

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The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. What is known?

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The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. What is known? Line-graphs of cubic graphs are normal [Patakfalvi 2008].

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. What is known? Line-graphs of cubic graphs are normal [Patakfalvi 2008]. Circulant graphs are normal [Wagler 2007].

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. What is known? Line-graphs of cubic graphs are normal [Patakfalvi 2008]. Circulant graphs are normal [Wagler 2007]. A few classes of sparse graphs have been show to be normal [Berry and Wagler 2013].

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. What is known? Line-graphs of cubic graphs are normal [Patakfalvi 2008]. Circulant graphs are normal [Wagler 2007]. A few classes of sparse graphs have been show to be normal [Berry and Wagler 2013]. Almost all d-regular graphs are normal when d is fixed [Hosseini, Mohar, Rezaei 2015].

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal.

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The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8 C5

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8 C5

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8 C5 C7

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8 C5 C7

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8 C5 C7 C7

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8 C5 C7 C7

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The Normal Graph Conjecture Sketch of the proof Conclusion

The Normal Graph Conjecture [De Simone, Körner 1999] A graph with no C5, C7 and C7 as an induced subgraph is normal. Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

In a graph of girth at least 8 C5 C7 C7

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The Normal Graph Conjecture Sketch of the proof Conclusion

The philosophy behind probabilistic arguments Take an object at random, and prove that with positive probability it satisfies the desired properties.

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The Normal Graph Conjecture Sketch of the proof Conclusion

The philosophy behind probabilistic arguments Take an object at random, and prove that with positive probability it satisfies the desired properties.

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The Normal Graph Conjecture Sketch of the proof Conclusion

The philosophy behind probabilistic arguments Take an object at random, and prove that with positive probability it satisfies the desired properties.

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The Normal Graph Conjecture Sketch of the proof Conclusion

The philosophy behind probabilistic arguments Take an object at random, and prove that with positive probability it satisfies the desired properties.

p

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The Normal Graph Conjecture Sketch of the proof Conclusion

The philosophy behind probabilistic arguments Take an object at random, and prove that with positive probability it satisfies the desired properties.

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The Normal Graph Conjecture Sketch of the proof Conclusion

The philosophy behind probabilistic arguments Take an object at random, and prove that with positive probability it satisfies the desired properties.

p

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The Normal Graph Conjecture Sketch of the proof Conclusion

The philosophy behind probabilistic arguments Take an object at random, and prove that with positive probability it satisfies the desired properties.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Properties We generate a random graph Gn,p with p = n−0.9. With good probability, we have the following properties:

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The Normal Graph Conjecture Sketch of the proof Conclusion

Properties We generate a random graph Gn,p with p = n−0.9. With good probability, we have the following properties: The number of cycles of length at most 7 is small.

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Properties We generate a random graph Gn,p with p = n−0.9. With good probability, we have the following properties: The number of cycles of length at most 7 is small. α(G) = o(n0.95).

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2. The graph induced by the edges of C is a spanning vertex-disjoint union of stars.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2. The graph induced by the edges of C is a spanning vertex-disjoint union of stars. Every member in S induces a stable set in G.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2. The graph induced by the edges of C is a spanning vertex-disjoint union of stars. Every member in S induces a stable set in G. C ∩ S = ∅ for every C ∈ C and S ∈ S.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2. The graph induced by the edges of C is a spanning vertex-disjoint union of stars. Every member in S induces a stable set in G. C ∩ S = ∅ for every C ∈ C and S ∈ S.

u v

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2. The graph induced by the edges of C is a spanning vertex-disjoint union of stars. Every member in S induces a stable set in G. C ∩ S = ∅ for every C ∈ C and S ∈ S.

u v

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2. The graph induced by the edges of C is a spanning vertex-disjoint union of stars. Every member in S induces a stable set in G. C ∩ S = ∅ for every C ∈ C and S ∈ S.

u v

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star covering Every member of C induces a clique K2 or K1 in G, where no K1 is included in some K2. The graph induced by the edges of C is a spanning vertex-disjoint union of stars. Every member in S induces a stable set in G. C ∩ S = ∅ for every C ∈ C and S ∈ S.

u v

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The Normal Graph Conjecture Sketch of the proof Conclusion

centers leaves

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centers leaves

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Key Lemma Stable sets are propagating through connected stars.

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Key Lemma Stable sets are propagating through connected stars.

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Key Lemma Stable sets are propagating through connected stars.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Key Lemma Stable sets are propagating through connected stars.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Key Lemma Stable sets are propagating through connected stars.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Key Lemma Stable sets are propagating through connected stars.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star system A star system (Q, S) of G is a spanning set of vertex disjoint stars with:

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Star system A star system (Q, S) of G is a spanning set of vertex disjoint stars with:

1 S is the set of stars. 16/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Star system A star system (Q, S) of G is a spanning set of vertex disjoint stars with:

1 S is the set of stars. 2 Q is the set of centers of the stars of S. 16/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Star system A star system (Q, S) of G is a spanning set of vertex disjoint stars with:

1 S is the set of stars. 2 Q is the set of centers of the stars of S. 16/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Star system A star system (Q, S) of G is a spanning set of vertex disjoint stars with:

1 S is the set of stars. 2 Q is the set of centers of the stars of S.

S1 S2 S3 Si ∈ S

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The Normal Graph Conjecture Sketch of the proof Conclusion

Star system A star system (Q, S) of G is a spanning set of vertex disjoint stars with:

1 S is the set of stars. 2 Q is the set of centers of the stars of S.

S1 S2 S3 x1 x2 x3 Si ∈ S xi ∈ Q

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The Normal Graph Conjecture Sketch of the proof Conclusion

Q∗ Given a star system (Q, S), we associate a directed graph Q∗ on vertex set Q and by letting xi → xj if a leaf of Si is adjacent to xj.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Q∗ Given a star system (Q, S), we associate a directed graph Q∗ on vertex set Q and by letting xi → xj if a leaf of Si is adjacent to xj.

xi xj xk

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The Normal Graph Conjecture Sketch of the proof Conclusion

Q∗ Given a star system (Q, S), we associate a directed graph Q∗ on vertex set Q and by letting xi → xj if a leaf of Si is adjacent to xj.

xi xj xk

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The Normal Graph Conjecture Sketch of the proof Conclusion

Q∗ Given a star system (Q, S), we associate a directed graph Q∗ on vertex set Q and by letting xi → xj if a leaf of Si is adjacent to xj.

xi xj xk

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The Normal Graph Conjecture Sketch of the proof Conclusion

Out-section A subset X of Q is an out-section if there exists v in Q such that for each x ∈ X, there exists a directed path in Q∗ from v to x.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Out-section A subset X of Q is an out-section if there exists v in Q such that for each x ∈ X, there exists a directed path in Q∗ from v to x.

x

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The Normal Graph Conjecture Sketch of the proof Conclusion

Out-section A subset X of Q is an out-section if there exists v in Q such that for each x ∈ X, there exists a directed path in Q∗ from v to x.

x v · · ·

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The Normal Graph Conjecture Sketch of the proof Conclusion

Out-section A subset X of Q is an out-section if there exists v in Q such that for each x ∈ X, there exists a directed path in Q∗ from v to x.

x v · · ·

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The Normal Graph Conjecture Sketch of the proof Conclusion

Out-section A subset X of Q is an out-section if there exists v in Q such that for each x ∈ X, there exists a directed path in Q∗ from v to x.

v x X

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The Normal Graph Conjecture Sketch of the proof Conclusion

Out-section A subset X of Q is an out-section if there exists v in Q such that for each x ∈ X, there exists a directed path in Q∗ from v to x.

v x X

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The Normal Graph Conjecture Sketch of the proof Conclusion

Out-section A subset X of Q is an out-section if there exists v in Q such that for each x ∈ X, there exists a directed path in Q∗ from v to x.

v x X

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The Normal Graph Conjecture Sketch of the proof Conclusion

Private neighbor Given a graph G and Q ⊆ V (G) partitioned into Q1, . . . , Q10, we say that w ∈ V (G) \ Q is a private neighbor of a vertex vi ∈ Qi if w is adjacent to vi but not to any vertex of Q1, . . . , Qi.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Private neighbor Given a graph G and Q ⊆ V (G) partitioned into Q1, . . . , Q10, we say that w ∈ V (G) \ Q is a private neighbor of a vertex vi ∈ Qi if w is adjacent to vi but not to any vertex of Q1, . . . , Qi.

Qi Qi+1 Qi−1 vi · · · · · ·

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The Normal Graph Conjecture Sketch of the proof Conclusion

Private neighbor Given a graph G and Q ⊆ V (G) partitioned into Q1, . . . , Q10, we say that w ∈ V (G) \ Q is a private neighbor of a vertex vi ∈ Qi if w is adjacent to vi but not to any vertex of Q1, . . . , Qi.

Qi Qi+1 Qi−1 vi w · · · · · ·

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The Normal Graph Conjecture Sketch of the proof Conclusion

Property JQ A graph G has the property JQ if for every choice of pairwise disjoint subsets of vertices J, Q1, . . . , Q10 with:

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The Normal Graph Conjecture Sketch of the proof Conclusion

Property JQ A graph G has the property JQ if for every choice of pairwise disjoint subsets of vertices J, Q1, . . . , Q10 with:

1 |J| ≤ n0.91 20/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Property JQ A graph G has the property JQ if for every choice of pairwise disjoint subsets of vertices J, Q1, . . . , Q10 with:

1 |J| ≤ n0.91 2

n0.9 1000 ≤ |Qi| ≤ n0.9 500 for all i ∈ {1, . . . , 10}

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The Normal Graph Conjecture Sketch of the proof Conclusion

Property JQ A graph G has the property JQ if for every choice of pairwise disjoint subsets of vertices J, Q1, . . . , Q10 with:

1 |J| ≤ n0.91 2

n0.9 1000 ≤ |Qi| ≤ n0.9 500 for all i ∈ {1, . . . , 10}

Then Q∗ over G \ J has an out-section whose set of private neighbors have size at least n0.95.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Lemma JQ P[G has the property JQ] = 1 − o(1).

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The Normal Graph Conjecture Sketch of the proof Conclusion

Lemma JQ P[G has the property JQ] = 1 − o(1). Proof Probabilistic arguments on Gn,p with p = n−9/10: Union bound. Markov’s bound. Chernoff’s bound.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Proof of the main theorem

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The Normal Graph Conjecture Sketch of the proof Conclusion

Proof of the main theorem Consider a random graph G = Gn,p with p = n−9/10.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Proof of the main theorem Consider a random graph G = Gn,p with p = n−9/10. For n sufficiently large, by the union bound and classical probabilistic arguments, there exists an n-vertex graph such that:

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The Normal Graph Conjecture Sketch of the proof Conclusion

Proof of the main theorem Consider a random graph G = Gn,p with p = n−9/10. For n sufficiently large, by the union bound and classical probabilistic arguments, there exists an n-vertex graph such that:

1 G has not too many small cycles. 22/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Proof of the main theorem Consider a random graph G = Gn,p with p = n−9/10. For n sufficiently large, by the union bound and classical probabilistic arguments, there exists an n-vertex graph such that:

1 G has not too many small cycles. 2 α(G) = o(n0.95). 22/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Proof of the main theorem Consider a random graph G = Gn,p with p = n−9/10. For n sufficiently large, by the union bound and classical probabilistic arguments, there exists an n-vertex graph such that:

1 G has not too many small cycles. 2 α(G) = o(n0.95). 3 G has property JQ. 22/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Consider a feedback vertex set S of the short cycles.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider a feedback vertex set S of the short cycles. Assume now for contradiction that G \ S is a normal graph.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider a feedback vertex set S of the short cycles. Assume now for contradiction that G \ S is a normal graph. Let S′ be the set of stars which have small size.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider a feedback vertex set S of the short cycles. Assume now for contradiction that G \ S is a normal graph. Let S′ be the set of stars which have small size. Consider now G \ (S ∪ S′).

G

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider a feedback vertex set S of the short cycles. Assume now for contradiction that G \ S is a normal graph. Let S′ be the set of stars which have small size. Consider now G \ (S ∪ S′).

G \ S

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider a feedback vertex set S of the short cycles. Assume now for contradiction that G \ S is a normal graph. Let S′ be the set of stars which have small size. Consider now G \ (S ∪ S′).

G \ S

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider a feedback vertex set S of the short cycles. Assume now for contradiction that G \ S is a normal graph. Let S′ be the set of stars which have small size. Consider now G \ (S ∪ S′). Let J = S ∪ S′.

G \ (S ∪ S′)

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider now Q∗ on Q based on the star covering of G \ J.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider now Q∗ on Q based on the star covering of G \ J.

G \ J

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider now Q∗ on Q based on the star covering of G \ J.

G \ J Q∗

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider now Q∗ on Q based on the star covering of G \ J.

Q∗

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The Normal Graph Conjecture Sketch of the proof Conclusion

Consider now Q∗ on Q based on the star covering of G \ J.

Q∗

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The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

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The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

1 |Q| > n0.9/3. 25/30

The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

1 |Q| > n0.9/3. 2 Every strongly connected component of Q∗ has size at most

n0.9/1000.

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The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

1 |Q| > n0.9/3. 2 Every strongly connected component of Q∗ has size at most

n0.9/1000. Let C1, . . . , Ck be the strongly connected components of Q∗ enumerated in a topological order.

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The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

1 |Q| > n0.9/3. 2 Every strongly connected component of Q∗ has size at most

n0.9/1000. Let C1, . . . , Ck be the strongly connected components of Q∗ enumerated in a topological order. Concatenate subsets of C1, . . . , Ck into blocks Q1, Q2, . . . , Q10 such that n0.9/1000 ≤ |Qi| ≤ n0.9/500.

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The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

1 |Q| > n0.9/3. 2 Every strongly connected component of Q∗ has size at most

n0.9/1000. Let C1, . . . , Ck be the strongly connected components of Q∗ enumerated in a topological order. Concatenate subsets of C1, . . . , Ck into blocks Q1, Q2, . . . , Q10 such that n0.9/1000 ≤ |Qi| ≤ n0.9/500.

· · · C1 C2 C3 C4 C5

It is possible because of

1 and 2 . 25/30

The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

1 |Q| > n0.9/3. 2 Every strongly connected component of Q∗ has size at most

n0.9/1000. Let C1, . . . , Ck be the strongly connected components of Q∗ enumerated in a topological order. Concatenate subsets of C1, . . . , Ck into blocks Q1, Q2, . . . , Q10 such that n0.9/1000 ≤ |Qi| ≤ n0.9/500.

· · · C1 C2 C3 C4 C5

It is possible because of

1 and 2 . 25/30

The Normal Graph Conjecture Sketch of the proof Conclusion

One can show that:

1 |Q| > n0.9/3. 2 Every strongly connected component of Q∗ has size at most

n0.9/1000. Let C1, . . . , Ck be the strongly connected components of Q∗ enumerated in a topological order. Concatenate subsets of C1, . . . , Ck into blocks Q1, Q2, . . . , Q10 such that n0.9/1000 ≤ |Qi| ≤ n0.9/500.

· · · C1 C2 C3 C4 C5 Q1 Q2

It is possible because of

1 and 2 . 25/30

The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi Qi−2 Qi−1

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi Qi−2 Qi−1

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi Qi−2 Qi−1 · · · Qj

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi Qi−2 Qi−1 · · · Qj xj

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi Qi−2 Qi−1 · · · Qj xj

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi Qi−2 Qi−1 · · · Qj xj

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The Normal Graph Conjecture Sketch of the proof Conclusion

Claim: If a vertex v of G \ (J ∪ Q) is a private neighbor of a vertex xi in Qi, then the edge xiv must be an edge of the clique covering. Proof:

xi v Qi Qi−2 Qi−1 · · · Qj xj

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The Normal Graph Conjecture Sketch of the proof Conclusion

By property JQ, we know that the private directed graph on stars formed by the private neighbors has an out-section O of size at least n0.95.

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The Normal Graph Conjecture Sketch of the proof Conclusion

By property JQ, we know that the private directed graph on stars formed by the private neighbors has an out-section O of size at least n0.95. Because the stars formed by private neighbors are in the clique covering, we can apply the stable set propagation lemma. Hence, we obtain an independent set of size at least n0.95.

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The Normal Graph Conjecture Sketch of the proof Conclusion

By property JQ, we know that the private directed graph on stars formed by the private neighbors has an out-section O of size at least n0.95. Because the stars formed by private neighbors are in the clique covering, we can apply the stable set propagation lemma. Hence, we obtain an independent set of size at least n0.95. Contradiction to α(G) = o(n0.95).

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following:

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following: G has a small number of short cycles.

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following: G has a small number of short cycles. G has a large number of connected stars.

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following: G has a small number of short cycles. G has a large number of connected stars. α(G) = o(n0.95).

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following: G has a small number of short cycles. G has a large number of connected stars. α(G) = o(n0.95). Let us remove short cycles.

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following: G has a small number of short cycles. G has a large number of connected stars. α(G) = o(n0.95). Let us remove short cycles. We have a graph of girth at least 8.

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following: G has a small number of short cycles. G has a large number of connected stars. α(G) = o(n0.95). Let us remove short cycles. We have a graph of girth at least 8. The large number of connected stars induces a stable set of size n0.95 in the star covering!

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The Normal Graph Conjecture Sketch of the proof Conclusion

In short? There exists an n-vertex graph G satisfying the following: G has a small number of short cycles. G has a large number of connected stars. α(G) = o(n0.95). Let us remove short cycles. We have a graph of girth at least 8. The large number of connected stars induces a stable set of size n0.95 in the star covering! Contradiction to the fact that α(G) = o(n0.95).

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The Normal Graph Conjecture Sketch of the proof Conclusion

Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Theorem [Harutyunyan, Pastor, Thomassé] There exists a graph G of girth at least 8 that is not normal. Counter-example to the Normal Graph Conjecture!

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The Normal Graph Conjecture Sketch of the proof Conclusion

Conclusion

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The Normal Graph Conjecture Sketch of the proof Conclusion

Conclusion Our counter-example is probabilistic. It might be interesting to look for a deterministic construction.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Conclusion Our counter-example is probabilistic. It might be interesting to look for a deterministic construction. Other classes of graphes might verify the conjecture.

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The Normal Graph Conjecture Sketch of the proof Conclusion

Conclusion Our counter-example is probabilistic. It might be interesting to look for a deterministic construction. Other classes of graphes might verify the conjecture. A good characterization of normal graphs in terms of graph theory?

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The Normal Graph Conjecture Sketch of the proof Conclusion

Conclusion Our counter-example is probabilistic. It might be interesting to look for a deterministic construction. Other classes of graphes might verify the conjecture. A good characterization of normal graphs in terms of graph theory? Thank you!

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