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

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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.

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The problem

Let G be a large graph and F a fixed one. Does G contain a “large” F-free subgraph? Question

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 2 / 9

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The problem

Let G be a d-regular graph (d large enough) and F a fixed graph. Does G contain a spanning F-free subgraph with large minimum degree? Question

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 2 / 9

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The problem

Let G be a d-regular graph (d large enough) and F a fixed graph. Does G contain a spanning F-free subgraph with large minimum degree? Question

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 2 / 9

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The problem

Let G be a d-regular graph (d large enough) and F a fixed graph. Does G contain a spanning F-free subgraph with large minimum degree? Question

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 2 / 9

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The problem

Let G be a d-regular graph (d large enough) and F a fixed graph. Does G contain a spanning F-free subgraph with large minimum degree? Question

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 2 / 9

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The problem

Let G be a d-regular graph (d large enough) and F a fixed graph. Does G contain a spanning F-free subgraph with large minimum degree? Question

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 2 / 9

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The problem

Let G be a d-regular graph (d large enough) and F a fixed graph. Does G contain a spanning F-free subgraph with large minimum degree? Question 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

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The conjecture

Turan numbers: ex(n, F) = max{e(G) : G subgraph of Kn and G is F-free} Since Kd+1 is d-regular we have, f (d, F) ≤ 2ex(d + 1, F) d + 1 = O ex(d, F) d

.

For every fixed graph F, f (d, F) = Θ ex(d, F) d

.

Conjecture (Foucaud, Krivelevich, P. (2013))

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 3 / 9

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Easy case: F = C3

Consider a partition V (G) = V1 ∪ V2 that maximizes e(V1, V2). Let H be the bipartite subgraph containing the edges E(V1, V2). Then, minimum degree of H is at least d/2, and H is C3-free. f (d, C3) = 1 2 + o(1)

d .

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 4 / 9

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General case: F with χ(F) = k

Consider a partition V (G) = V1 ∪ · · · ∪ Vk−1 that maximizes

i=j e(Vi, Vj).

Let H be the (k − 1)-partite subgraph containing the edges ∪i=jE(Vi, Vj). Then, minimum degree of H is at least

1 −

1 k−1

d, and

H is F-free. f (d, F) =

1 −

1 k − 1 + o(1)

d .

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 5 / 9

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General case: F with χ(F) = k

Consider a partition V (G) = V1 ∪ · · · ∪ Vk−1 that maximizes

i=j e(Vi, Vj).

Let H be the (k − 1)-partite subgraph containing the edges ∪i=jE(Vi, Vj). Then, minimum degree of H is at least

1 −

1 k−1

d, and

H is F-free. f (d, F) =

1 −

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

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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) .

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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 = C4 is a cycle of length 4. Then ex(d, C4) = O(d3/2) and we have the upper bound ex(d, C4) = O( √ d) . If d is large, f (d, C4) = Ω

d1/3

. Theorem (Kun (2013)) If d is large, f (d, C4) = Ω √ d log d

.

Theorem (Foucaud, Krivelevich, P. (2013))

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 6 / 9

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The theorem

If d is large, f (d, C4) = Θ √ d

.

Theorem (P., Reed (2014))

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 7 / 9

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The theorem

If d is large, f (d, C4) = Θ √ d

.

Theorem (P., Reed (2014)) Is the regularity condition needed? Essentially yes. Let G = Kδ,∆ (max degree ∆ \ min degree δ). Any C4-free subgraph H ⊆ Kδ,∆ has minimum degree δH = O

δ

√ ∆

/. If δ ≪ ∆, then δ(H) = o(

√ δ).

Guillem Perarnau Spanning F-free subgraphs with large minimum degree 7 / 9

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