On the structure of Dense graphs with bounded clique number

Oberkampf, Heiner; Schacht, Mathias

doi:10.1017/s0963548319000324

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…One may ask then, with respect to the above discussion, whether we can replace the property of having bounded chromatic number with the property of admitting a homomorphism to a graph of bounded order with additional properties. This question was posed by Thomassen in the specific case of triangle‐free graphs, and motivated Oberkampf and Schacht to introduce the homomorphism threshold

δ_{prefixhom} false(H false)

of a graph H :

\begin{matrix} true \\ right δ_{hom} (H) & = & left trueprefixinf false{d : \exists C = C false(H, d false) s.t. \forall G \in prefixForb false(H, d false) \\ left \exists G^{'} \in {Forb}_{C} (H) s.t. G is homomorphic to G^{'}} . \end{matrix}

In words,

δ_{prefixhom} false(H false)

is the infimum over all

d \in [0, 1]

such that every H ‐free graph with n vertices and minimum degree at least

d n

is homomorphic to an H ‐free graph of bounded order (independent of n ). Note that the definition of

δ_{prefixhom} false(H false)

extends naturally if we replace H by a family

scriptH

of graphs.…”

Section: Introductionsupporting

confidence: 54%

“…This result was extended by Goddard and Lyle to

K_{r}

‐free graphs for

r \geq 4

, and, in particular, we know that

δ_{prefixhom} false(K_{r} false) = δ_{χ} false(K_{r} false) = \frac{2 r - 5}{2 r - 3}

. Oberkampf and Schacht gave a new proof of this result avoiding the Regularity Lemma (which was used in Łuczak's proof), and asked for the determination of the homomorphism threshold of the odd cycle,

δ_{prefixhom} false(C_{2 ℓ - 1} false)

, and

δ_{prefixhom} false({C_{3}, \dots, C_{2 ℓ - 1}} false)

for

ℓ \geq 3

. As our first main result, we determine the value of the second of these two parameters in the case

ℓ = 3

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The homomorphism threshold of ‐free graphs

Letzter

Snyder

2018

Journal of Graph Theory

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We determine the structure of {C3,C5}‐free graphs with n vertices and minimum degree larger than n/5: such graphs are homomorphic to the graph obtained from a (5k−3)‐cycle by adding all chords of length 1(mod5), for some k. This answers a question of Messuti and Schacht. We deduce that the homomorphism threshold of {C3,C5}‐free graphs is 1/5, thus answering a question of Oberkampf and Schacht.

show abstract

δ_{prefixhom} false(H false)

of a graph H :

\begin{matrix} true \\ right δ_{hom} (H) & = & left trueprefixinf false{d : \exists C = C false(H, d false) s.t. \forall G \in prefixForb false(H, d false) \\ left \exists G^{'} \in {Forb}_{C} (H) s.t. G is homomorphic to G^{'}} . \end{matrix}

In words,

δ_{prefixhom} false(H false)

is the infimum over all

d \in [0, 1]

such that every H ‐free graph with n vertices and minimum degree at least

d n

is homomorphic to an H ‐free graph of bounded order (independent of n ). Note that the definition of

δ_{prefixhom} false(H false)

extends naturally if we replace H by a family

scriptH

of graphs.…”

Section: Introductionsupporting

confidence: 54%

“…This result was extended by Goddard and Lyle to

K_{r}

‐free graphs for

r \geq 4

, and, in particular, we know that

δ_{prefixhom} false(K_{r} false) = δ_{χ} false(K_{r} false) = \frac{2 r - 5}{2 r - 3}

δ_{prefixhom} false(C_{2 ℓ - 1} false)

, and

δ_{prefixhom} false({C_{3}, \dots, C_{2 ℓ - 1}} false)

for

ℓ \geq 3

. As our first main result, we determine the value of the second of these two parameters in the case

ℓ = 3

.…”

Section: Introductionmentioning

confidence: 99%

The homomorphism threshold of ‐free graphs

Letzter

Snyder

2018

Journal of Graph Theory

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show abstract

“…In other words, the chromatic threshold measures what minimum degree conditions force an

{\rm{ {\mathcal F} }}

‐free graph to have a homomorphism to a graph of bounded order, and the homomorphism threshold further requires this bounded graph to itself be

{\rm{ {\mathcal F} }}

‐free. Due to the efforts of many researchers [12, 23, 29, 31, 32, 34], it is now known that

{\delta }_{\chi }({K}_{r})={\delta }_{\text{hom}}({K}_{r})=\frac{2r-5}{2r-3}.

…”

Section: Discussionmentioning

confidence: 99%

“…[0, 1] : there exists > 0 such that ( ) for all ( , )} χ and the homomorphism threshold of  is In other words, the chromatic threshold measures what minimum degree conditions force an  -free graph to have a homomorphism to a graph of bounded order, and the homomorphism threshold further requires this bounded graph to itself be  -free. Due to the efforts of many researchers [12,23,29,31,32,34], it is now known that Conjecture 4.9. There exists a graph H for which…”

mentioning

confidence: 99%

Minimum degree and the graph removal lemma

Fox

Wigderson

2022

Journal of Graph Theory

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The clique removal lemma says that for every r ≥ 3 $r\ge 3$ and ε > 0 $\varepsilon \gt 0$, there exists some δ > 0 $\delta \gt 0$ so that every n $n$‐vertex graph G $G$ with fewer than δ n r $\delta {n}^{r}$ copies of K r ${K}_{r}$ can be made K r ${K}_{r}$‐free by removing at most ε n 2 $\varepsilon {n}^{2}$ edges. The dependence of δ $\delta $ on ε $\varepsilon $ in this result is notoriously difficult to determine: it is known that δ − 1 ${\delta }^{-1}$ must be at least super‐polynomial in ε − 1 ${\varepsilon }^{-1}$, and that it is at most of tower type in log ε − 1 $\mathrm{log} {\varepsilon }^{-1}$. We prove that if one imposes an appropriate minimum degree condition on G $G$, then one can actually take δ $\delta $ to be a linear function of ε $\varepsilon $ in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super‐polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.

show abstract

“…In other words, the chromatic threshold measures what minimum degree conditions force an F -free graph to have a homomorphism to a graph of bounded order, and the homomorphism threshold further requires this bounded graph to itself be F -free. Due to the efforts of many researchers [12,23,29,31,32,34], it is now known that δ χ (K r ) = δ hom (K r ) = 2r − 5 2r − 3 .…”

Section: The Chromatic and Homomorphism Thresholdsmentioning

confidence: 99%

Minimum degree and the graph removal lemma

Fox¹,

Wigderson²

2021

Preprint

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The clique removal lemma says that for every r ≥ 3 and ε > 0, there exists some δ > 0 so that every n-vertex graph G with fewer than δn r copies of K r can be made K r -free by removing at most εn 2 edges. The dependence of δ on ε in this result is notoriously difficult to determine: it is known that δ −1 must be at least superpolynomial in ε −1 , and that it is at most of tower type in log ε −1 .We prove that if one imposes an appropriate minimum degree condition on G, then one can actually take δ to be a linear function of ε in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super-polynomial, as in the unrestricted removal lemma.We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma.

show abstract

On the structure of Dense graphs with bounded clique number

Cited by 6 publications

References 12 publications

The homomorphism threshold of ‐free graphs

The homomorphism threshold of ‐free graphs

Minimum degree and the graph removal lemma

Minimum degree and the graph removal lemma

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