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1. Pure Pairs. V. Excluding Some Long Subdivision

   https://doi.org/10.1007/s00493-023-00025-8  

   Scott, Alex; Seymour, Paul; Spirkl, Sophie (June 2023, Combinatorica)

   A “pure pair” in a graph G is a pair A, B of disjoint subsets of V(G) such that A is complete or anticomplete to B. Jacob Fox showed that for all ε>0, there is a comparability graph G with n vertices, where n is large, in which there is no pure pair A, B with |A|,|B|≥εn. He also proved that for all c>0 there exists ε>0 such that for every comparability graph G with n>1 vertices, there is a pure pair A, B with |A|,|B|≥εn1−c; and conjectured that the same holds for every perfect graph G. We prove this conjecture and strengthen it in several ways. In particular, we show that for all c>0, and all ℓ1,ℓ2≥4/c+9, there exists ε>0 such that, if G is an (n>1)-vertex graph with no hole of length exactly ℓ1 and no antihole of length exactly ℓ2, then there is a pure pair A, B in G with |A|≥εn and |B|≥εn1−c. This is further strengthened, replacing excluding a hole by excluding some “long” subdivision of a general graph. 

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2. Planar Lattice Subsets with Minimal Vertex Boundary

   https://doi.org/10.37236/10055  

   Gupta, Radhika; Levcovitz, Ivan; Margolis, Alexander; Stark, Emily (July 2021, The Electronic Journal of Combinatorics)

   A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $$X$$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $$X$$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $$X$$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $$G$$ of minimal sets whose vertices are congruence classes of minimal sets of $$X$$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected. 

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3. Unitary signings and induced subgraphs of Cayley graphs of $$\mathbb{Z}_2^{n}$$

   https://doi.org/10.19086/aic.17912  

   Alon, Noga; Zheng, Kai (November 2020, Advances in Combinatorics)

   null (Ed.)

   Boolean functions play an important role in many different areas of computer science. The _local sensitivity_ of a Boolean function $$f:\{0,1\}^n\to \{0,1\}$$ on an input $$x\in\{0,1\}^n$$ is the number of coordinates whose flip changes the value of $f(x)$, i.e., the number of i's such that $$f(x)\not=f(x+e_i)$$, where $$e_i$$ is the $$i$$-th unit vector. The _sensitivity_ of a Boolean function is its maximum local sensitivity. In other words, the sensitivity measures the robustness of a Boolean function with respect to a perturbation of its input. Another notion that measures the robustness is block sensitivity. The _local block sensitivity_ of a Boolean function $$f:\{0,1\}^n\to \{0,1\}$ on an input $$x\in\{0,1\}^n$$ is the number of disjoint subsets $$I$$ of $$\{1,..,n\}$$ such that flipping the coordinates indexed by $$I$$ changes the value of $f(x)$$, and the _block sensitivity_ of $$f$$ is its maximum local block sensitivity. Since the local block sensitivity is at least the local sensitivity for any input $$x$$, the block sensitivity of $$f$$ is at least the sensitivity of $$f$$.The next example demonstrates that the block sensitivity of a Boolean function is not linearly bounded by its sensitivity. Fix an integer $$k\ge 2$$ and define a Boolean function $$f:\{0,1\}^{2k^2}\to\{0,1\}$$ as follows: the coordinates of $$x\in\{0,1\}^{2k^2}$$ are split into $$k$$ blocks of size $2k$ each and $f(x)=1$ if and only if at least one of the blocks contains exactly two entries equal to one and these entries are consecutive. While the sensitivity of the function $$f$$ is $2k$, its block sensitivity is $k^2$. The Sensitivity Conjecture, made by Nisan and Szegedy in 1992, asserts that the block sensitivity of a Boolean function is polynomially bounded by its sensivity. The example above shows that the degree of such a polynomial must be at least two.The Sensitivity Conjecture has been recently proven by Huang in [Annals of Mathematics 190 (2019), 949-955](https://doi.org/10.4007/annals.2019.190.3.6). He proved the following combinatorial statement that implies the conjecture (with the degree of the polynomial equal to four): any subset of more than half of the vertices of the $$n$$-dimensional cube $$\{0,1\}^n$$ induces a subgraph that contains a vertex with degree at least $$\sqrt{n}$$. The present article extends this result as follows: every Cayley graph with the vertex set $$\{0,1\}^n$$ and any generating set of size $$d$$ (the vertex set is viewed as a vector space over the binary field) satisfies that any subset of more than half of its vertices induces a subgraph that contains a vertex of degree at least $$\sqrt{d}$$. In particular, when the generating set consists of the $$n$$ unit vectors, the Cayley graph is the $$n$$-dimensional hypercube. 

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4. Concatenating Bipartite Graphs

   https://doi.org/10.37236/8451  

   Chudnovsky, Maria; Hompe, Patrick; Scott, Alex; Seymour, Paul; Spirkl, Sophie (April 2022, The Electronic Journal of Combinatorics)

   Let $$x,y\in (0,1]$$, and let $A,B,C$ be disjoint nonempty stable subsets of a graph $$G$$, where every vertex in $$A$$ has at least $x|B|$ neighbours in $$B$$, and every vertex in $$B$$ has at least $y|C|$ neighbours in $$C$$, and there are no edges between $A,C$. We denote by $$\phi(x,y)$$ the maximum $$z$$ such that, in all such graphs $$G$$, there is a vertex $$v\in C$$ that is joined to at least $z|A|$ vertices in $$A$$ by two-edge paths. This function has some interesting properties: we show, for instance, that $$\phi(x,y)=\phi(y,x)$$ for all $x,y$, and there is a discontinuity in $$\phi(x,x)$$ when $1/x$ is an integer. For $z=1/2, 2/3,1/3,3/4,2/5,3/5$, we try to find the (complicated) boundary between the set of pairs $(x,y)$ with $$\phi(x,y)\ge z$$ and the pairs with $$\phi(x,y)1/3$. 

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5. Subdivided Claws and the Clique-Stable Set Separation Property

   https://doi.org/10.1007/978-3-030-62497-2  

   Chudnovsky, Maria and (February 2021, MATRIX book series)

   null (Ed.)

   Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c ∈ N such that for every graph G ∈ C there is a collection P of partitions (X, Y ) of the vertex set of G with |P| ≤ |V (G)| c and with the following property: if K is a clique of G, and S is a stable set of G, and K ∩ S = ∅, then there is (X, Y ) ∈ P with K ⊆ X and S ⊆ Y . In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by M. G¨o¨os in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S, K of graphs and show that for every S ∈ S and K ∈ K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property. 

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