Spanning F -free subgraphs with large minimum degree Guillem - PowerPoint PPT Presentation

Spanning F -free subgraphs with large minimum degree Guillem Perarnau SIAM Conference on Discrete Math, Minneapolis - June 19th, 2014 McGill University, Montreal, Canada joint work with Bruce Reed.

The problem Let G be a large graph and F a fixed one. Question Does G contain a “large” F -free subgraph? Guillem Perarnau Spanning F -free subgraphs with large minimum degree 2 / 9

The problem Let G be a d -regular graph ( d large enough) and F a fixed graph. Question Does G contain a spanning F -free subgraph with large minimum degree? Guillem Perarnau Spanning F -free subgraphs with large minimum degree 2 / 9

The problem Let G be a d -regular graph ( d large enough) and F a fixed graph. Question Does G contain a spanning F -free subgraph with large minimum degree? Guillem Perarnau Spanning F -free subgraphs with large minimum degree 2 / 9

The problem Let G be a d -regular graph ( d large enough) and F a fixed graph. Question Does G contain a spanning F -free subgraph with large minimum degree? Guillem Perarnau Spanning F -free subgraphs with large minimum degree 2 / 9

The problem Let G be a d -regular graph ( d large enough) and F a fixed graph. Question Does G contain a spanning F -free subgraph with large minimum degree? Guillem Perarnau Spanning F -free subgraphs with large minimum degree 2 / 9

The problem Let G be a d -regular graph ( d large enough) and F a fixed graph. Question Does G contain a spanning F -free subgraph with large minimum degree? Guillem Perarnau Spanning F -free subgraphs with large minimum degree 2 / 9

The problem Let G be a d -regular graph ( d large enough) and F a fixed graph. Question Does G contain a spanning F -free subgraph with large minimum degree? f ( d , F ) := max { t : for every d -reg graph G there exists a spanning F -free subgraph with minimum degree ≥ t } Guillem Perarnau Spanning F -free subgraphs with large minimum degree 2 / 9

The conjecture Turan numbers: ex ( n , F ) = max { e ( G ) : G subgraph of K n and G is F -free } Since K d +1 is d -regular we have, � ex ( d , F ) � f ( d , F ) ≤ 2 ex ( d + 1 , F ) = O . d + 1 d Conjecture (Foucaud, Krivelevich, P. (2013)) For every fixed graph F , � ex ( d , F ) � f ( d , F ) = Θ . d Guillem Perarnau Spanning F -free subgraphs with large minimum degree 3 / 9

Easy case: F = C 3 Consider a partition V ( G ) = V 1 ∪ V 2 that maximizes e ( V 1 , V 2 ). Let H be the bipartite subgraph containing the edges E ( V 1 , V 2 ). Then, minimum degree of H is at least d / 2, and H is C 3 -free. � 1 � f ( d , C 3 ) = 2 + o (1) d . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 4 / 9

General case: F with χ ( F ) = k Consider a partition V ( G ) = V 1 ∪ · · · ∪ V k − 1 that maximizes � i � = j e ( V i , V j ). Let H be the ( k − 1)-partite subgraph containing the edges ∪ i � = j E ( V i , V j ). Then, minimum degree of H is at least � � 1 1 − d , and k − 1 H is F -free. � 1 � f ( d , F ) = 1 − k − 1 + o (1) d . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 5 / 9

General case: F with χ ( F ) = k Consider a partition V ( G ) = V 1 ∪ · · · ∪ V k − 1 that maximizes � i � = j e ( V i , V j ). Let H be the ( k − 1)-partite subgraph containing the edges ∪ i � = j E ( V i , V j ). Then, minimum degree of H is at least � � 1 1 − d , and k − 1 H is F -free. � 1 � f ( d , F ) = 1 − k − 1 + o (1) d . That solves completely the case when χ ( F ) ≥ 3. If χ ( F ) = 2 ( F bipartite), f ( d , F ) = o ( d ) . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 5 / 9

Bipartite case: F with χ ( F ) = 2 Simplest case: F = T is a tree. Then ex ( d , T ) = O ( d ) and the conjecture states ex ( d , T ) = Θ(1) .

Bipartite case: F with χ ( F ) = 2 Simplest case: F = T is a tree. Then ex ( d , T ) = O ( d ) and the conjecture states ex ( d , T ) = Θ(1) . Simplest non-trivial case: F = C 4 is a cycle of length 4. Then ex ( d , C 4 ) = O ( d 3 / 2 ) and we have the upper bound √ ex ( d , C 4 ) = O ( d ) . Theorem (Kun (2013)) If d is large, � d 1 / 3 � f ( d , C 4 ) = Ω . Theorem (Foucaud, Krivelevich, P. (2013)) If d is large, � √ � d f ( d , C 4 ) = Ω . log d Guillem Perarnau Spanning F -free subgraphs with large minimum degree 6 / 9

The theorem Theorem (P., Reed (2014)) If d is large, � √ � f ( d , C 4 ) = Θ d . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 7 / 9

The theorem Theorem (P., Reed (2014)) If d is large, � √ � f ( d , C 4 ) = Θ d . Is the regularity condition needed? Essentially yes . Let G = K δ, ∆ (max degree ∆ \ min degree δ ). Any C 4 -free subgraph H ⊆ K δ, ∆ √ � � δ has minimum degree δ H = O /. If δ ≪ ∆, then δ ( H ) = o ( δ ). √ ∆ Guillem Perarnau Spanning F -free subgraphs with large minimum degree 7 / 9

Drawing the proof Bipartize graph G : still minimum degree ≥ d / 2. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Randomly color A Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof For a vertex a ∈ A keep color if . . . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof For a vertex a ∈ A keep color if . . . . . . not many bad neighbors. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Then also remove dangerous edges. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof For a vertex a ∈ A uncolor if . . . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof For a vertex a ∈ A uncolor if . . . . . . too many bad neighbors. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Then, uncolor and keep all the edges Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof After first iteration we get a partial coloring. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof We keep iterating . . . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof We keep iterating . . . . . . until a small number of vertices are uncolored. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Then, color all them at once. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Important Properties: 1.- for every v ∈ B , N ( v ) is rainbow. 2.- the minimum degree is Ω( d ). Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Color B in the same way. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Consider a extremal graph G without 4-cycles ( V ( G ) = colors). Use the coloring on G to embed it onto G Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Keep just the edges of G that agree with edges in G . Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Fact I: because of the embedding, no rainbow 4-cycles. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof Fact I: because of the embedding, no rainbow 4-cycles Fact II: because of the properties of the coloring, no non -rainbow 4-cycles. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

Drawing the proof The subgraph obtained is C 4 -free and has large minimum degree. Guillem Perarnau Spanning F -free subgraphs with large minimum degree 8 / 9

The real theorem Let F = { F 1 , . . . , F s } be a family of fixed graphs. We say that F is closed if for every F ∈ F and G G is F -free ⇐ ⇒ no locally injective homomorphism from F to G . Theorem (P., Reed (2014)) Let F be a closed family and d large, � ex ( d , F ) � f ( d , F ) = Θ . d Examples: cycles: F = { C 3 , . . . , C 2 r +1 } (existence of subgraphs with large girth and large minimum degree), F = { C 2 p : p prime } . complete bipartite graphs: for any a i , b i , i ≤ n , F = ∪ i K a i , b i . First unknown case: C 8 Guillem Perarnau Spanning F -free subgraphs with large minimum degree 9 / 9

The real theorem Let F = { F 1 , . . . , F s } be a family of fixed graphs. We say that F is closed if for every F ∈ F and G G is F -free ⇐ ⇒ no locally injective homomorphism from F to G . Theorem (P., Reed (2014)) Let F be a closed family and d large, � ex ( d , F ) � f ( d , F ) = Θ . d Examples: cycles: F = { C 3 , . . . , C 2 r +1 } (existence of subgraphs with large girth and large minimum degree), F = { C 2 p : p prime } . complete bipartite graphs: for any a i , b i , i ≤ n , F = ∪ i K a i , b i . First unknown case: C 8 THANKS FOR YOUR ATTENTION Guillem Perarnau Spanning F -free subgraphs with large minimum degree 9 / 9