Doubly-critical hypergraphs Tom´ aˇ s Kaiser Department of Mathematics and Institute for Theoretical Computer Science University of West Bohemia Pilsen, Czech Republic ık and R. ˇ based on joint work with M. Stehl´ Skrekovski (and J.-S. Sereni) GT2015, Nyborg, August 27, 2015 Tom´ aˇ s Kaiser Doubly-critical hypergraphs
The doubly-critical graph conjecture A graph G is doubly-critical if for every edge xy of G , χ ( G − x − y ) = χ ( G ) − 2. Conjecture (Erd˝ os, Lov´ asz 1966) The only (connected) doubly-critical graphs are the complete graphs. Tom´ aˇ s Kaiser Doubly-critical hypergraphs
The Tihany conjecture The preceding is a special case of the Tihany conjecture of Erd˝ os and Lov´ asz: Conjecture Let a , b ≥ 2 . If G is a graph with ω ( G ) < χ ( G ) = a + b − 1 , then V ( G ) can be partitioned as A ∪ B, where χ ( G [ A ]) ≥ a and χ ( G [ B ]) ≥ b. Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Doubly-critical graphs with small χ Special cases of the doubly-critical conjecture for k -chromatic graphs with k small: case k = 3 trivial, k = 4 easy Theorem (Stiebitz, 1987) The only 5-doubly-critical graph is K 5 . Partial results are due to Krusenstjerna-Hafstrom and Toft / Kawarabayashi, Pedersen and Toft / Albar and Gon¸ calves: if k ≥ 6, then G is 6-connected, if 6 ≤ k ≤ 8, then G is contractible to K k . Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Doubly-critical graphs with small χ Special cases of the doubly-critical conjecture for k -chromatic graphs with k small: case k = 3 trivial, k = 4 easy Theorem (Stiebitz, 1987) The only 5-doubly-critical graph is K 5 . Partial results are due to Krusenstjerna-Hafstrom and Toft / Kawarabayashi, Pedersen and Toft / Albar and Gon¸ calves: if k ≥ 6, then G is 6-connected, if 6 ≤ k ≤ 8, then G is contractible to K k . Tom´ aˇ s Kaiser Doubly-critical hypergraphs
A topological view a complex on a set V = family of subsets of V (faces), closed under taking subsets consider the independence complex I ( G ) of a k -chromatic graph G with vertex set V let [ V ] denote the complex on V whose faces are all possible subsets of V for any complex K on V and an integer α , define α · K to be the complex on V whose faces are F 1 ∪ · · · ∪ F α , where each F i is a face of K if G is k -chromatic, then ( k − 1) · I ( G ) � = [ V ] = k · I ( G ) G is doubly-critical (and k -chromatic) iff ( k − 2) · I ( G ) is the ‘Alexander dual’ of I ( G ) maybe we can characterise such complexes using topological tools? Tom´ aˇ s Kaiser Doubly-critical hypergraphs
A topological view a complex on a set V = family of subsets of V (faces), closed under taking subsets consider the independence complex I ( G ) of a k -chromatic graph G with vertex set V let [ V ] denote the complex on V whose faces are all possible subsets of V for any complex K on V and an integer α , define α · K to be the complex on V whose faces are F 1 ∪ · · · ∪ F α , where each F i is a face of K if G is k -chromatic, then ( k − 1) · I ( G ) � = [ V ] = k · I ( G ) G is doubly-critical (and k -chromatic) iff ( k − 2) · I ( G ) is the ‘Alexander dual’ of I ( G ) maybe we can characterise such complexes using topological tools? Tom´ aˇ s Kaiser Doubly-critical hypergraphs
A topological view a complex on a set V = family of subsets of V (faces), closed under taking subsets consider the independence complex I ( G ) of a k -chromatic graph G with vertex set V let [ V ] denote the complex on V whose faces are all possible subsets of V for any complex K on V and an integer α , define α · K to be the complex on V whose faces are F 1 ∪ · · · ∪ F α , where each F i is a face of K if G is k -chromatic, then ( k − 1) · I ( G ) � = [ V ] = k · I ( G ) G is doubly-critical (and k -chromatic) iff ( k − 2) · I ( G ) is the ‘Alexander dual’ of I ( G ) maybe we can characterise such complexes using topological tools? Tom´ aˇ s Kaiser Doubly-critical hypergraphs
A topological view a complex on a set V = family of subsets of V (faces), closed under taking subsets consider the independence complex I ( G ) of a k -chromatic graph G with vertex set V let [ V ] denote the complex on V whose faces are all possible subsets of V for any complex K on V and an integer α , define α · K to be the complex on V whose faces are F 1 ∪ · · · ∪ F α , where each F i is a face of K if G is k -chromatic, then ( k − 1) · I ( G ) � = [ V ] = k · I ( G ) G is doubly-critical (and k -chromatic) iff ( k − 2) · I ( G ) is the ‘Alexander dual’ of I ( G ) maybe we can characterise such complexes using topological tools? Tom´ aˇ s Kaiser Doubly-critical hypergraphs
From complexes to hypergraphs The property of complexes we are interested in is the following (let’s call such complexes interesting): 1 we need, say, k faces of K to cover V 2 but if we remove any (minimal) non-face from V , k − 2 faces suffice to cover the rest But this is equivalent to hypergraph colouring: let C ( K ) be the hypergraph on V whose hyperedges are all minimal non-faces (circuits) of K for a hyperedge e of a hypergraph H , let H \ \ e be obtained by removing the vertices of e and all hyperedges intersecting e Observation K is interesting iff for every hyperedge e of C ( K ) , χ ( C ( K ) \ \ e ) = χ ( C ( K )) − 2 . Tom´ aˇ s Kaiser Doubly-critical hypergraphs
From complexes to hypergraphs The property of complexes we are interested in is the following (let’s call such complexes interesting): 1 we need, say, k faces of K to cover V 2 but if we remove any (minimal) non-face from V , k − 2 faces suffice to cover the rest But this is equivalent to hypergraph colouring: let C ( K ) be the hypergraph on V whose hyperedges are all minimal non-faces (circuits) of K for a hyperedge e of a hypergraph H , let H \ \ e be obtained by removing the vertices of e and all hyperedges intersecting e Observation K is interesting iff for every hyperedge e of C ( K ) , χ ( C ( K ) \ \ e ) = χ ( C ( K )) − 2 . Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Doubly-critical hypergraphs Definition A hypergraph H is doubly-critical if for every hyperedge e , χ ( H \ \ e ) = χ ( H ) − 2 . can we hope for a characterisation? Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Examples of doubly-critical hypergraphs adding isolated vertices adding a vertex and joining it completely by 2-edges the complete k + 1-uniform hypergraph on ak + 1 vertices (where a , k ≥ 1) Tom´ aˇ s Kaiser Doubly-critical hypergraphs
The bad news Observation Every intersecting 3 -chromatic hypergraph is doubly-critical. constructions yielding infinite classes of (uniform) examples given in a 1975 paper of Erd˝ os and Lov´ asz no hope for a characterisation Tom´ aˇ s Kaiser Doubly-critical hypergraphs
The bad news Observation Every intersecting 3 -chromatic hypergraph is doubly-critical. constructions yielding infinite classes of (uniform) examples given in a 1975 paper of Erd˝ os and Lov´ asz no hope for a characterisation Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Good news: matroids ık, ˇ Theorem (TK, Stehl´ Skrekovski) If H is the hypergraph of circuits of a matroid, then H is doubly-critical if and only if H is a uniform matroid U r ar +1 for some a , r ≥ 1 , plus possibly some coloops. Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Proof sketch Assume M is a matroid on V , H = C ( M ) is doubly-critical and χ ( H ) = k . Theorem (Edmonds) V can be covered by ℓ independent sets iff for each X ⊆ V , | X | ≤ ℓ · rk ( X ) . since V cannot be covered by k − 1 independent sets, there is a set X with | X | ≥ ( k − 1) r + 1, where r = rk ( X ) if we remove any circuit C from M , we can cover by k − 2 independent sets, so | X \ C | ≤ ( k − 2) · rk ( X \ C ) ≤ ( k − 2) r thus | X ∩ C | ≥ r + 1 for any C we infer that the independent sets of M are the sets intersecting X in at most r elements; the theorem follows (with a = k − 1) Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Proof sketch Assume M is a matroid on V , H = C ( M ) is doubly-critical and χ ( H ) = k . Theorem (Edmonds) V can be covered by ℓ independent sets iff for each X ⊆ V , | X | ≤ ℓ · rk ( X ) . since V cannot be covered by k − 1 independent sets, there is a set X with | X | ≥ ( k − 1) r + 1, where r = rk ( X ) if we remove any circuit C from M , we can cover by k − 2 independent sets, so | X \ C | ≤ ( k − 2) · rk ( X \ C ) ≤ ( k − 2) r thus | X ∩ C | ≥ r + 1 for any C we infer that the independent sets of M are the sets intersecting X in at most r elements; the theorem follows (with a = k − 1) Tom´ aˇ s Kaiser Doubly-critical hypergraphs
Proof sketch Assume M is a matroid on V , H = C ( M ) is doubly-critical and χ ( H ) = k . Theorem (Edmonds) V can be covered by ℓ independent sets iff for each X ⊆ V , | X | ≤ ℓ · rk ( X ) . since V cannot be covered by k − 1 independent sets, there is a set X with | X | ≥ ( k − 1) r + 1, where r = rk ( X ) if we remove any circuit C from M , we can cover by k − 2 independent sets, so | X \ C | ≤ ( k − 2) · rk ( X \ C ) ≤ ( k − 2) r thus | X ∩ C | ≥ r + 1 for any C we infer that the independent sets of M are the sets intersecting X in at most r elements; the theorem follows (with a = k − 1) Tom´ aˇ s Kaiser Doubly-critical hypergraphs
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