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

The Normal Graph Conjecture Sketch of the proof Conclusion The Normal Graph Conjecture [ De Simone, Körner 1999 ] A graph with no C 5 , C 7 and C 7 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 C 5 10/30

The Normal Graph Conjecture Sketch of the proof Conclusion The Normal Graph Conjecture [ De Simone, Körner 1999 ] A graph with no C 5 , C 7 and C 7 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 C 5 10/30

The Normal Graph Conjecture Sketch of the proof Conclusion The Normal Graph Conjecture [ De Simone, Körner 1999 ] A graph with no C 5 , C 7 and C 7 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 C 5 C 7 10/30

The Normal Graph Conjecture Sketch of the proof Conclusion The Normal Graph Conjecture [ De Simone, Körner 1999 ] A graph with no C 5 , C 7 and C 7 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 C 5 C 7 10/30

The Normal Graph Conjecture Sketch of the proof Conclusion The Normal Graph Conjecture [ De Simone, Körner 1999 ] A graph with no C 5 , C 7 and C 7 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 C 5 C 7 C 7 10/30

The Normal Graph Conjecture Sketch of the proof Conclusion The Normal Graph Conjecture [ De Simone, Körner 1999 ] A graph with no C 5 , C 7 and C 7 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 C 5 C 7 C 7 10/30

The Normal Graph Conjecture Sketch of the proof Conclusion The Normal Graph Conjecture [ De Simone, Körner 1999 ] A graph with no C 5 , C 7 and C 7 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 C 5 C 7 C 7 10/30

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. 11/30

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. 11/30

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. 11/30

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 11/30

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. 11/30

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 11/30

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. 11/30

The Normal Graph Conjecture Sketch of the proof Conclusion Properties We generate a random graph G n , p with p = n − 0 . 9 . With good probability, we have the following properties: 12/30

The Normal Graph Conjecture Sketch of the proof Conclusion Properties We generate a random graph G n , 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 . 12/30

The Normal Graph Conjecture Sketch of the proof Conclusion Properties We generate a random graph G n , 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 ( n 0 . 95 ). 12/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . The graph induced by the edges of C is a spanning vertex-disjoint union of stars. 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . 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 . 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . 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 . 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . 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 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . 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 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . 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 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion Star covering Every member of C induces a clique K 2 or K 1 in G , where no K 1 is included in some K 2 . 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 13/30

The Normal Graph Conjecture Sketch of the proof Conclusion leaves centers 14/30

The Normal Graph Conjecture Sketch of the proof Conclusion leaves centers 14/30

The Normal Graph Conjecture Sketch of the proof Conclusion Key Lemma Stable sets are propagating through connected stars. 15/30

The Normal Graph Conjecture Sketch of the proof Conclusion Key Lemma Stable sets are propagating through connected stars. 15/30

The Normal Graph Conjecture Sketch of the proof Conclusion Key Lemma Stable sets are propagating through connected stars. 15/30

The Normal Graph Conjecture Sketch of the proof Conclusion Key Lemma Stable sets are propagating through connected stars. 15/30

The Normal Graph Conjecture Sketch of the proof Conclusion Key Lemma Stable sets are propagating through connected stars. 15/30

The Normal Graph Conjecture Sketch of the proof Conclusion Key Lemma Stable sets are propagating through connected stars. 15/30

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: 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. 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 . S 1 S 2 S 3 S i ∈ 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 . S 1 S 2 S 3 x 1 x 2 x 3 S i ∈ S x i ∈ Q 16/30

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 x i → x j if a leaf of S i is adjacent to x j . 17/30

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 x i → x j if a leaf of S i is adjacent to x j . x i x j x k 17/30

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 x i → x j if a leaf of S i is adjacent to x j . x i x j x k 17/30

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 x i → x j if a leaf of S i is adjacent to x j . x i x j x k 17/30

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 . 18/30

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 18/30

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 18/30

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 18/30

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 x 18/30

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 x 18/30

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 x 18/30

The Normal Graph Conjecture Sketch of the proof Conclusion Private neighbor Given a graph G and Q ⊆ V ( G ) partitioned into Q 1 , . . . , Q 10 , we say that w ∈ V ( G ) \ Q is a private neighbor of a vertex v i ∈ Q i if w is adjacent to v i but not to any vertex of Q 1 , . . . , Q i . 19/30

The Normal Graph Conjecture Sketch of the proof Conclusion Private neighbor Given a graph G and Q ⊆ V ( G ) partitioned into Q 1 , . . . , Q 10 , we say that w ∈ V ( G ) \ Q is a private neighbor of a vertex v i ∈ Q i if w is adjacent to v i but not to any vertex of Q 1 , . . . , Q i . · · · · · · v i Q i − 1 Q i Q i +1 19/30

The Normal Graph Conjecture Sketch of the proof Conclusion Private neighbor Given a graph G and Q ⊆ V ( G ) partitioned into Q 1 , . . . , Q 10 , we say that w ∈ V ( G ) \ Q is a private neighbor of a vertex v i ∈ Q i if w is adjacent to v i but not to any vertex of Q 1 , . . . , Q i . w · · · · · · v i Q i − 1 Q i Q i +1 19/30

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 , Q 1 , . . . , Q 10 with: 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 , Q 1 , . . . , Q 10 with: 1 | J | ≤ n 0 . 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 , Q 1 , . . . , Q 10 with: 1 | J | ≤ n 0 . 91 1000 ≤ | Q i | ≤ n 0 . 9 n 0 . 9 500 for all i ∈ { 1 , . . . , 10 } 2 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 , Q 1 , . . . , Q 10 with: 1 | J | ≤ n 0 . 91 1000 ≤ | Q i | ≤ n 0 . 9 n 0 . 9 500 for all i ∈ { 1 , . . . , 10 } 2 Then Q ∗ over G \ J has an out-section whose set of private neighbors have size at least n 0 . 95 . 20/30

The Normal Graph Conjecture Sketch of the proof Conclusion Lemma JQ P [ G has the property JQ ] = 1 − o (1) . 21/30

The Normal Graph Conjecture Sketch of the proof Conclusion Lemma JQ P [ G has the property JQ ] = 1 − o (1) . Proof Probabilistic arguments on G n , p with p = n − 9 / 10 : Union bound. Markov’s bound. Chernoff’s bound. 21/30

The Normal Graph Conjecture Sketch of the proof Conclusion Proof of the main theorem 22/30

The Normal Graph Conjecture Sketch of the proof Conclusion Proof of the main theorem Consider a random graph G = G n , p with p = n − 9 / 10 . 22/30

The Normal Graph Conjecture Sketch of the proof Conclusion Proof of the main theorem Consider a random graph G = G n , 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: 22/30

The Normal Graph Conjecture Sketch of the proof Conclusion Proof of the main theorem Consider a random graph G = G n , 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 = G n , 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 ( n 0 . 95 ). 22/30

The Normal Graph Conjecture Sketch of the proof Conclusion Proof of the main theorem Consider a random graph G = G n , 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 ( n 0 . 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. 23/30

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. 23/30