Doubly-critical hypergraphs Tom a s Kaiser Department of - - PowerPoint PPT Presentation

Doubly-critical hypergraphs

Tom´ aˇ s Kaiser

Department of Mathematics and Institute for Theoretical Computer Science University of West Bohemia Pilsen, Czech Republic

based on joint work with M. Stehl´ ık and R. ˇ 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˝

• s, 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˝

• s

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 K5. 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 Kk.

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 K5. 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 Kk.

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 F1 ∪ · · · ∪ Fα, where each Fi 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 F1 ∪ · · · ∪ Fα, where each Fi 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 F1 ∪ · · · ∪ Fα, where each Fi 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 F1 ∪ · · · ∪ Fα, where each Fi 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˝

• s 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˝

• s and Lov´

asz no hope for a characterisation

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Good news: matroids

Theorem (TK, Stehl´ ık, ˇ 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 Ur

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

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

Back to graphs

if G is any graph [without isolated vertices], we define DC(G) (the double criticality subgraph of G) as a spanning subgraph of G such that: an edge xy of G is in DC(G) (is ‘good’) if χ(G − x − y) = χ(G) − 2.

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Degrees in DC(G)

If we restrict to graphs G with connected complement, DC(G) becomes a lot sparser. Small examples suggest:

1 DC(G) might always contain a vertex of degree 0, or at least 2 each vertex of G might be incident with an edge not in

DC(G) Unfortunately, both (1) and (2) are false, as shown by the complement of: Conjecture If G is k-chromatic with connected complement, then DC(G) contains a vertex of degree at most k − 2. [Maybe even k − 4 — this holds for k = 4 and in small 5- and 6-critical graphs.]

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Degrees in DC(G)

If we restrict to graphs G with connected complement, DC(G) becomes a lot sparser. Small examples suggest:

1 DC(G) might always contain a vertex of degree 0, or at least 2 each vertex of G might be incident with an edge not in

DC(G) Unfortunately, both (1) and (2) are false, as shown by the complement of: Conjecture If G is k-chromatic with connected complement, then DC(G) contains a vertex of degree at most k − 2. [Maybe even k − 4 — this holds for k = 4 and in small 5- and 6-critical graphs.]

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Degrees in DC(G)

If we restrict to graphs G with connected complement, DC(G) becomes a lot sparser. Small examples suggest:

1 DC(G) might always contain a vertex of degree 0, or at least 2 each vertex of G might be incident with an edge not in

DC(G) Unfortunately, both (1) and (2) are false, as shown by the complement of: Conjecture If G is k-chromatic with connected complement, then DC(G) contains a vertex of degree at most k − 2. [Maybe even k − 4 — this holds for k = 4 and in small 5- and 6-critical graphs.]

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Size of DC(G)

Theorem (Kawarabayashi et al.) If G is 4-critical and incomplete, then the ratio of the number of edges of DC(G) and the number of edges of G is at most 1/2. The value 1/2 is attained if and only if G is an odd wheel. Conjecture (Kawarabayashi et al.) If G is 5-critical incomplete and has n vertices and m edges, then DC(G) has at most (2 + 1 3n − 5) · m 3

• edges. Moreover, it has this many if and only if it is an odd wheel

with an added universal vertex.

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Size of DC(G)

Theorem (Kawarabayashi et al.) If G is 4-critical and incomplete, then the ratio of the number of edges of DC(G) and the number of edges of G is at most 1/2. The value 1/2 is attained if and only if G is an odd wheel. Conjecture (Kawarabayashi et al.) If G is 5-critical incomplete and has n vertices and m edges, then DC(G) has at most (2 + 1 3n − 5) · m 3

• edges. Moreover, it has this many if and only if it is an odd wheel

with an added universal vertex.

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Claw-free graphs

Problem Are there incomplete doubly-critical claw-free graphs (without isolated vertices)? Known for various classes of claw-free graphs: line graphs of multigraphs (Kostochka and Stiebitz) quasi-line graphs and graphs with α ≤ 2 (Balogh et al.) {claw, K5 − e}-free graphs and {claw, 5-wheel}-free graphs (easy) the case k = 7 not hard to verify Theorem (Chudnovsky, Fradkin and Plumettaz 2013+) If G is a claw-free graph with χ(G) > ω(G), then there is a clique K with |K| ≤ 5 such that χ(G − K) > χ(G) − |K|.

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Claw-free graphs

Problem Are there incomplete doubly-critical claw-free graphs (without isolated vertices)? Known for various classes of claw-free graphs: line graphs of multigraphs (Kostochka and Stiebitz) quasi-line graphs and graphs with α ≤ 2 (Balogh et al.) {claw, K5 − e}-free graphs and {claw, 5-wheel}-free graphs (easy) the case k = 7 not hard to verify Theorem (Chudnovsky, Fradkin and Plumettaz 2013+) If G is a claw-free graph with χ(G) > ω(G), then there is a clique K with |K| ≤ 5 such that χ(G − K) > χ(G) − |K|.

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Claw-free graphs

Problem Are there incomplete doubly-critical claw-free graphs (without isolated vertices)? Known for various classes of claw-free graphs: line graphs of multigraphs (Kostochka and Stiebitz) quasi-line graphs and graphs with α ≤ 2 (Balogh et al.) {claw, K5 − e}-free graphs and {claw, 5-wheel}-free graphs (easy) the case k = 7 not hard to verify Theorem (Chudnovsky, Fradkin and Plumettaz 2013+) If G is a claw-free graph with χ(G) > ω(G), then there is a clique K with |K| ≤ 5 such that χ(G − K) > χ(G) − |K|.

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Edgeless DC(G)

For some classes of claw-free graphs, DC(G) can be shown to have no edges: if G is quasi-line or α(G) = 2 (implicit in Balogh et al.) if G is {claw, 5-wheel}-free if G is {claw, K5 − e}-free There are, however, claw-free graphs G such that DC(G) has edges: e.g., complements of

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Edgeless DC(G)

For some classes of claw-free graphs, DC(G) can be shown to have no edges: if G is quasi-line or α(G) = 2 (implicit in Balogh et al.) if G is {claw, 5-wheel}-free if G is {claw, K5 − e}-free There are, however, claw-free graphs G such that DC(G) has edges: e.g., complements of

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

A last conjecture on DC(G)

Conjecture (KSS) If G is any graph, then DC(G) is perfect. In particular: Conjecture Assume that G contains an induced 5-cycle C. Then at least one edge of C is not an edge of DC(G).

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

A last conjecture on DC(G)

Conjecture (KSS) If G is any graph, then DC(G) is perfect. In particular: Conjecture Assume that G contains an induced 5-cycle C. Then at least one edge of C is not an edge of DC(G).

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Thank you for your attention . . . and. . . Best wishes to Bjarne!

Tom´ aˇ s Kaiser Doubly-critical hypergraphs

Thank you for your attention . . . and. . . Best wishes to Bjarne!

Tom´ aˇ s Kaiser Doubly-critical hypergraphs