Embedding trees in dense graphs
Václav Rozhoň
February 14, 2019
joint work with T. Klimošová and D. Piguet
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 1 / 11 Alternative definition: substructures in graphs
Extremal graph theory
Definition (Extremal graph theory, Bollobás 1976) Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 2 / 11 Extremal graph theory
Definition (Extremal graph theory, Bollobás 1976) Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians.
Alternative definition: substructures in graphs
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 2 / 11 Density of edges vs. density of triangles (Razborov 2008)
(image from the book of Lovász)
Extremal graph theory
Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2/4 edges then it contains a triangle.
Generalisations?
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 3 / 11 Extremal graph theory
Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2/4 edges then it contains a triangle.
Generalisations? Density of edges vs. density of triangles (Razborov 2008)
(image from the book of Lovász)
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 3 / 11 3/2 The answer for C4 is of order n , lower bound via finite projective planes. What is the answer for trees?
Extremal graph theory
Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2/4 edges then it contains a triangle.
Generalisations? Other cliques (Turán 1941), asymptotically for all non-bipartite graphs (Erd˝os-Stone 1946)
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 4 / 11 What is the answer for trees?
Extremal graph theory
Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2/4 edges then it contains a triangle.
Generalisations? Other cliques (Turán 1941), asymptotically for all non-bipartite graphs (Erd˝os-Stone 1946) 3/2 The answer for C4 is of order n , lower bound via finite projective planes.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 4 / 11 Extremal graph theory
Theorem (Mantel 1907) Graph G has n vertices. If G has more than n2/4 edges then it contains a triangle.
Generalisations? Other cliques (Turán 1941), asymptotically for all non-bipartite graphs (Erd˝os-Stone 1946) 3/2 The answer for C4 is of order n , lower bound via finite projective planes. What is the answer for trees?
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 4 / 11 Average degree of 2k suffices. Conjecture (Erdos-Sós)˝ Any graph with average degree greater than k 2 contains any tree on k vertices as a subgraph. −
Partial results: special trees (paths – Erd˝os, Gallai 1959)
special graphs (without C4 – Saclé, Wozniak 1997) n and k differ by constant (Görlich, Zak˙ 2016)
Erdos-Sós˝ conjecture
Fix any tree T on k vertices. There are graphs with average degree k 2 that do not contain T . −
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 5 / 11 Conjecture (Erdos-Sós)˝ Any graph with average degree greater than k 2 contains any tree on k vertices as a subgraph. −
Partial results: special trees (paths – Erd˝os, Gallai 1959)
special graphs (without C4 – Saclé, Wozniak 1997) n and k differ by constant (Görlich, Zak˙ 2016)
Erdos-Sós˝ conjecture
Fix any tree T on k vertices. There are graphs with average degree k 2 that do not contain T . − Average degree of 2k suffices.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 5 / 11 Partial results: special trees (paths – Erd˝os, Gallai 1959)
special graphs (without C4 – Saclé, Wozniak 1997) n and k differ by constant (Görlich, Zak˙ 2016)
Erdos-Sós˝ conjecture
Fix any tree T on k vertices. There are graphs with average degree k 2 that do not contain T . − Average degree of 2k suffices. Conjecture (Erdos-Sós)˝ Any graph with average degree greater than k 2 contains any tree on k vertices as a subgraph. −
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 5 / 11 Erdos-Sós˝ conjecture
Fix any tree T on k vertices. There are graphs with average degree k 2 that do not contain T . − Average degree of 2k suffices. Conjecture (Erdos-Sós)˝ Any graph with average degree greater than k 2 contains any tree on k vertices as a subgraph. −
Partial results: special trees (paths – Erd˝os, Gallai 1959)
special graphs (without C4 – Saclé, Wozniak 1997) n and k differ by constant (Görlich, Zak˙ 2016)
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 5 / 11 One can get reasonably close if the size of the tree is comparable with the size of the graph. Below ∆ is maximum degree and deg average degree. Theorem (R. 2019), also (Besomi, Pavez-Signé, Stein 2019+) Let be a class of trees such that T : ∆(T ) o( T ). ThenT any graph G with deg(G) = ∀T +∈o T( G ) contains∈ | any| T . | | | | ∈ T
Erdos-Sós˝ conjecture
Theorem (announced by Ajtai, Komlós, Simonovits, Szemerédi)
The conjecture holds for k k0. ≥
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 6 / 11 Erdos-Sós˝ conjecture
Theorem (announced by Ajtai, Komlós, Simonovits, Szemerédi)
The conjecture holds for k k0. ≥ One can get reasonably close if the size of the tree is comparable with the size of the graph. Below ∆ is maximum degree and deg average degree. Theorem (R. 2019), also (Besomi, Pavez-Signé, Stein 2019+) Let be a class of trees such that T : ∆(T ) o( T ). ThenT any graph G with deg(G) = ∀T +∈o T( G ) contains∈ | any| T . | | | | ∈ T
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 6 / 11 Long strand of results of increasing strength: 1 + ε approximation by Hladký, Komlós, Piguet, Simonovits, Stein, Szemerédi from 2017. Conjecture ( Simonovits) If at least rn vertices of G have degree at least k, then G contains any tree with k + 1 vertices and at most r(k + 1) vertices in one colour class as a subgraph.
Loebl-Komlós-Sós
Conjecture (Loebl, Komlós, Sós 1995) If at least n/2 vertices of G have degree at least k, then G contains any tree with k + 1 vertices as a subgraph.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 7 / 11 Conjecture ( Simonovits) If at least rn vertices of G have degree at least k, then G contains any tree with k + 1 vertices and at most r(k + 1) vertices in one colour class as a subgraph.
Loebl-Komlós-Sós
Conjecture (Loebl, Komlós, Sós 1995) If at least n/2 vertices of G have degree at least k, then G contains any tree with k + 1 vertices as a subgraph.
Long strand of results of increasing strength: 1 + ε approximation by Hladký, Komlós, Piguet, Simonovits, Stein, Szemerédi from 2017.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 7 / 11 Loebl-Komlós-Sós
Conjecture (Loebl, Komlós, Sós 1995) If at least n/2 vertices of G have degree at least k, then G contains any tree with k + 1 vertices as a subgraph.
Long strand of results of increasing strength: 1 + ε approximation by Hladký, Komlós, Piguet, Simonovits, Stein, Szemerédi from 2017. Conjecture ( Simonovits) If at least rn vertices of G have degree at least k, then G contains any tree with k + 1 vertices and at most r(k + 1) vertices in one colour class as a subgraph.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 7 / 11 Turns out that one can get this ’r’ trade-off also in the proof of previous Erd˝os-Sós result.
Loebl-Komlós-Sós
Conjecture (Loebl, Komlós, Sós, Simonovits) If at least rn vertices of G have degree at least k, then G contains any tree with k + 1 vertices and at most r(k + 1) vertices in one colour class as a subgraph.
Theorem (Klimošová, Piguet, R. 2019+) If at least rn vertices of G have degree at least k+o(n), then G contains any tree with k + 1 vertices and at most r(k + 1) vertices in one colour class as a subgraph.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 8 / 11 Loebl-Komlós-Sós
Conjecture (Loebl, Komlós, Sós, Simonovits) If at least rn vertices of G have degree at least k, then G contains any tree with k + 1 vertices and at most r(k + 1) vertices in one colour class as a subgraph.
Theorem (Klimošová, Piguet, R. 2019+) If at least rn vertices of G have degree at least k+o(n), then G contains any tree with k + 1 vertices and at most r(k + 1) vertices in one colour class as a subgraph.
Turns out that one can get this ’r’ trade-off also in the proof of previous Erd˝os-Sós result.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 8 / 11 (Pseudo)random graphs have good expansion. Szemerédi regularity lemma: dense graph = cluster graph + pseudorandomness (but we have to pay ε fraction of edges). This enables us to embed any small subtree (ε0n for very small ε0). Aim is to find suitable decomposition of the tree such that small subtrees are embedded by Szemerédi and the macro structure by us. We reduced the problem to a certain fractional variant of itself. But now we have much simpler tree structure to work with.
General technique
Expansion of the host graph can compensate for the lack of degree.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 9 / 11 Szemerédi regularity lemma: dense graph = cluster graph + pseudorandomness (but we have to pay ε fraction of edges). This enables us to embed any small subtree (ε0n for very small ε0). Aim is to find suitable decomposition of the tree such that small subtrees are embedded by Szemerédi and the macro structure by us. We reduced the problem to a certain fractional variant of itself. But now we have much simpler tree structure to work with.
General technique
Expansion of the host graph can compensate for the lack of degree. (Pseudo)random graphs have good expansion.
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 9 / 11 This enables us to embed any small subtree (ε0n for very small ε0). Aim is to find suitable decomposition of the tree such that small subtrees are embedded by Szemerédi and the macro structure by us. We reduced the problem to a certain fractional variant of itself. But now we have much simpler tree structure to work with.
General technique
Expansion of the host graph can compensate for the lack of degree. (Pseudo)random graphs have good expansion. Szemerédi regularity lemma: dense graph = cluster graph + pseudorandomness (but we have to pay ε fraction of edges).
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 9 / 11 Aim is to find suitable decomposition of the tree such that small subtrees are embedded by Szemerédi and the macro structure by us. We reduced the problem to a certain fractional variant of itself. But now we have much simpler tree structure to work with.
General technique
Expansion of the host graph can compensate for the lack of degree. (Pseudo)random graphs have good expansion. Szemerédi regularity lemma: dense graph = cluster graph + pseudorandomness (but we have to pay ε fraction of edges). This enables us to embed any small subtree (ε0n for very small ε0).
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 9 / 11 We reduced the problem to a certain fractional variant of itself. But now we have much simpler tree structure to work with.
General technique
Expansion of the host graph can compensate for the lack of degree. (Pseudo)random graphs have good expansion. Szemerédi regularity lemma: dense graph = cluster graph + pseudorandomness (but we have to pay ε fraction of edges). This enables us to embed any small subtree (ε0n for very small ε0). Aim is to find suitable decomposition of the tree such that small subtrees are embedded by
Szemerédi and the macro structure by us. 3 2
1 5 3 2 3 5
2 2 1 3 2 1
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 9 / 11 General technique
Expansion of the host graph can compensate for the lack of degree. (Pseudo)random graphs have good expansion. Szemerédi regularity lemma: dense graph = cluster graph + pseudorandomness (but we have to pay ε fraction of edges). This enables us to embed any small subtree (ε0n for very small ε0). Aim is to find suitable decomposition of the tree such that small subtrees are embedded by
Szemerédi and the macro structure by us. 3 2
We reduced the problem to a certain fractional 1 5 3 2 variant of itself. But now we have much 3 5
simpler tree structure to work with. 2 2 1 3 2 1
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 9 / 11 Proof of the Erdos-Sós˝ result
Proof. Condition on the maximum degree actually gives even simpler decomposition. After decomposition of G and T look at a high degree cluster of G and a maximal matching in its neighbourhood. Provide (almost) greedy algorithm for embedding.
WA WB
v1
DA DB
M1
O1
M2 W
O2 D0
00 D (former WB)
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 10 / 11 Proof of the Loebl-Komlós-Sós result
Proof. After decomposition of G and T ’discharge’ into several configurations, embedding for each one being straightforward.
S1 SM S1 SM S1 SM
p p y a100 a10 a100 a10 a100 a10
1 2 1 2 1 2
3 3 a a a 20 y 20 y 20 a1000 x b1 x b1 x 4 b1 a200 a200 a200
1 r 1 r x v L : deg(v ,S S ) a − x v L : deg(v ,S S ) a − x v L : deg(v ,S ) = 0 3 → 1 ∈ 1 M ∪ 1 ≥ 2 · r L → 1 ∈ 1 M ∪ 1 ≥ 2 · r L → 1 ∈ 1 0 L y v L : deg(v ,L) rk y v L : deg(v ,L) rk y v S : deg(v ) rk → 2 ∈ 2 ≥ → 2 ∈ 2 ≥ → 2 ∈ 2 ≥
S1 SM S1 SM
1 r b − 1 · r a100 a10 a10 1 r b0 − 1 · r
2 1 2 3 3 4 1 1 a a0 b 20 y a1000 2 10 y a1000 x b1 x b100 a200
x v L : deg(v ,S ) = 0 x v L∗ → 1 ∈ 1 0 L → 1 ∈ A L y v : deg(v ,L) b y v : deg(v ,L) b → 2 2 ≥ 1 → 2 2 ≥ 1
Václav Rozhoň Embedding trees in dense graphs February 14, 2019 11 / 11