Showing papers in "Electronic Journal of Combinatorics in 2021"
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TL;DR: The answer to two fundamental questions posed by Kaul and Mudrock is that the answer to both these questions is yes, and is shown to be yes even if the authors require $p=1$.
Abstract: DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořak and Postle. The chromatic polynomial of a graph is a notion that has been extensively studied since the early 20th century. The chromatic polynomial of graph $G$ is denoted $P(G,m)$, and it is equal to the number of proper $m$-colorings of $G$. In 2019, Kaul and Mudrock introduced an analogue of the chromatic polynomial for DP-coloring; specifically, the DP color function of graph $G$ is denoted $P_{DP}(G,m)$. Two fundamental questions posed by Kaul and Mudrock are: (1) For any graph $G$ with $n$ vertices, is it the case that $P(G,m)-P_{DP}(G,m) = O(m^{n-3})$ as $m \rightarrow \infty$? and (2) For every graph $G$, does there exist $p,N \in \mathbb{N}$ such that $P_{DP}(K_p \vee G, m) = P(K_p \vee G, m)$ whenever $m \geq N$? We show that the answer to both these questions is yes. In fact, we show the answer to (2) is yes even if we require $p=1$.
14 citations
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TL;DR: The powergraph is always perfect; and the groups whose power graph is a threshold graph are determined completely.
Abstract: Funding: CSIR, India (Grant No-09/983(0037)/2019-EMR-I), SERB, India through Core Research Grant (File Number-CRG/2020/000447).
13 citations
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TL;DR: This paper introduces a vertex-weighted version of the Tutte symmetric function XB and shows that this function admits a deletion-contraction relation, and gives several new methods for constructing nonisomorphic graphs with equal chromatic asymmetric function.
Abstract: This paper has two main parts. First, we consider the Tutte symmetric function XB, a generalization of the chromatic symmetric function. We introduce a vertex-weighted version of XB and show that this function admits a deletion-contraction relation. We also demonstrate that the vertex-weighted XB admits spanning-tree and spanning-forest expansions generalizing those of the Tutte polynomial by connecting XB to other graph functions. Second, we give several methods for constructing nonisomorphic graphs with equal chromatic and Tutte symmetric functions, and use them to provide specific examples.
13 citations
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TL;DR: This work proposes as a conjecture a simple characterization of finite sets F of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of F as an induce subdigraph.
Abstract: The dichromatic number of a digraph D is the minimum number of colors needed to color its vertices in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has been a recent center of study. In this work we look at possible extensions of Gyarfas-Sumner conjecture. More precisely, we propose as a conjecture a simple characterization of finite sets F of digraphs such that every oriented graph with sufficiently large dichromatic number must contain a member of F as an induce subdigraph. Among notable results, we prove that oriented triangle-free graphs without a directed path of length 3 are 2-colorable. If condition of "triangle-free" is replaced with "K 4-free", then we have an upper bound of 414. We also show that an orientation of complete multipartite graph with no directed triangle is 2-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.
12 citations
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TL;DR: It is shown that for any permutations sortable by $k$ the set is characterised by finitely many patterns, answering a question of Claesson and Gu{\dh}mundsson.
Abstract: We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $k\in\mathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Gu{\dh}mundsson.
Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called \emph{$2$-avoidance}.
12 citations
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TL;DR: As a generalization of birational promotion on a product of two chains, birational Coxeter-motion is introduced, and it is proved that it enjoys periodicity and file homomesy.
Abstract: Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistic. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and prove that it enjoys periodicity and file homomesy.
10 citations
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TL;DR: This note extends the result by Bohman, Frieze, and Martin on the threshold in $\mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$, and discusses embeddings of bounded degree trees and other spanning structures in this model.
Abstract: In the model of randomly perturbed graphs we consider the union of a deterministic graph $\mathcal{G}_\alpha$ with minimum degree $\alpha n$ and the binomial random graph $\mathbb{G}(n,p)$. This model was introduced by Bohman, Frieze, and Martin and for Hamilton cycles their result bridges the gap between Dirac's theorem and the results by Posa and Korsunov on the threshold in $\mathbb{G}(n,p)$. In this note we extend this result in $\mathcal{G}_\alpha \cup \mathbb{G}(n,p)$ to sparser graphs with $\alpha=o(1)$. More precisely, for any $\varepsilon>0$ and $\alpha \colon \mathbb{N} \mapsto (0,1)$ we show that a.a.s. $\mathcal{G}_\alpha \cup \mathbb{G}(n,\beta /n)$ is Hamiltonian, where $\beta = -(6 + \varepsilon) \log(\alpha)$. If $\alpha>0$ is a fixed constant this gives the aforementioned result by Bohman, Frieze, and Martin and if $\alpha=O(1/n)$ the random part $\mathbb{G}(n,p)$ is sufficient for a Hamilton cycle. We also discuss embeddings of bounded degree trees and other spanning structures in this model, which lead to interesting questions on almost spanning embeddings into $\mathbb{G}(n,p)$.
10 citations
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TL;DR: The classification of complete multipartite graphs whose edge rings are nearly Gorenstein was studied in this paper, as well as the classification of finite perfect graphs whose stable set rings are near the edge rings.
Abstract: The classification of complete multipartite graphs whose edge rings are nearly Gorenstein as well as that of finite perfect graphs whose stable set rings are nearly Gorenstein is achieved.
9 citations
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TL;DR: In this article, the authors studied the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts.
Abstract: We study the connection between triangulations of a type $A$ root polytope and the resonance arrangement, a hyperplane arrangement that shows up in a surprising number of contexts. Despite an elementary definition for the resonance arrangement, the number of resonance chambers has only been computed up to the $n=8$ dimensional case. We focus on data structures for labeling chambers, such as sign vectors and sets of alternating trees, with an aim at better understanding the structure of the resonance arrangement, and, in particular, enumerating its chambers. Along the way, we make connections with similar (and similarly difficult) enumeration questions. With the root polytope viewpoint, we relate resonance chambers to the chambers of polynomiality of the Kostant partition function. With the hyperplane viewpoint, we clarify the connections between resonance chambers and threshold functions. In particular, we show that the base-2 logarithm of the number of resonance chambers is asymptotically $n^2$.
9 citations
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TL;DR: It is proved that some sufficient conditions for a graph G are established in order to guarantee that its line graph L(G) has the Perfect-Matching-Hamiltonian property.
Abstract: A graph admitting a perfect matching has the Perfect-Matching-Hamiltonian property (for short the PMH-property) if each of its perfect matchings can be extended to a Hamiltonian cycle. In this paper we establish some sufficient conditions for a graph $G$ in order to guarantee that its line graph $L(G)$ has the PMH-property. In particular, we prove that this happens when $G$ is (i) a Hamiltonian graph with maximum degree at most $3$, (ii) a complete graph, or (iii) an arbitrarily traceable graph. Further related questions and open problems are proposed along the paper.
8 citations
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TL;DR: In this paper, a signed graph is defined as a pair (G, σ) where G is a graph and σ : E(G) → {+, −} is a signature which assigns to each edge of G a sign.
Abstract: A signed graph is a pair (G, σ), where G is a graph and σ : E(G) → {+, −} is a signature which assigns to each edge of G a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph (G, σ) a circular r-coloring of (G, σ) is an assignment ψ of points of a circle of circumference r to the vertices of G such that for every edge e = uv of G, if σ(e) = +, then ψ(u) and ψ(v) have distance at least 1, and if σ(e) = −, then ψ(v) and the antipodal of ψ(u) have distance at least 1. The circular chromatic number χ c (G, σ) of a signed graph (G, σ) is the infimum of those r for which (G, σ) admits a circular r-coloring. For a graph G, we define the signed circular chromatic number of G to be max{χ c (G, σ) : σ is a signature of G}. We study basic properties of circular coloring of signed graphs and develop tools for calculating χ c (G, σ). We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular, we determine the supremum of the signed circular chromatic number of k-chromatic graphs of large girth, of simple bipartite planar graphs, d-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is 4 + 2 3. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Macajova, Raspaud, and Skoviera.
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TL;DR: In this article, it was shown that all moments of the k-cut number of conditioned Galton-Watson trees converges after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees.
Abstract: The $k$-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the k-cut number of conditioned Galton-Watson trees converges after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the k-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.
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TL;DR: The coordinate ring of an $L$-convex polyomino is studied and its regularity is determined in terms of the maximal number of rooks that can be placed in thepolyomino.
Abstract: We study the coordinate ring of an L-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein L-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen-Macaulay type of any L-convex polyomino in terms of the maximal rectangles covering it. Though the main results are of algebraic nature, all proofs are combinatorial.
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TL;DR: This paper proves that among graphs with $m$ edges and maximum degree at most $r$ the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliquesof size $r+ 1$ together with the colex graph using the remainder of the edges.
Abstract: Recently Cutler and Radcliffe proved that the graph on $n$ vertices with maximum degree at most $r$ having the most cliques is a disjoint union of $\lfloor n/(r+1)\rfloor$ cliques of size $r+1$ together with a clique on the remainder of the vertices. It is very natural also to consider this question when the limiting resource is edges rather than vertices. In this paper we prove that among graphs with $m$ edges and maximum degree at most $r$, the graph that has the most cliques of size at least two is the disjoint union of $\bigl\lfloor m \bigm/\binom{r+1}{2} \bigr\rfloor$ cliques of size $r+1$ together with the colex graph using the remainder of the edges.
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TL;DR: The question of whether there is an n-satisfactory coloring is equivalent to a problem about tilings, and this is used to give a geometric characterization of multiplicative colorings.
Abstract: For which values of $n$ can we color the positive integers with precisely $n$ colors in such a way that for any $a$, the numbers $a,2a,\dots,na$ all get different colors? Pach posed the question around 2008-9. Particular cases appeared in KoMaL in April 2010, and the general version appeared in May 2010 on MathOverflow, posted by Palvolgyi. The question remains open. We discuss the known partial results and investigate a series of related matters attempting to understand the structure of these $n$-satisfactory colorings.
Specifically, we show that there is an $n$-satisfactory coloring whenever there is an abelian group operation $\oplus$ on the set $\{1,2,\dots,n\}$ compatible with multiplication in the sense that whenever $i$, $j$ and $ij$ are in $\{1,\dots,n\}$, then $ij=i\oplus j$. This includes in particular the cases where $n+1$ is prime, or $2n+1$ is prime, or $n=p^2-p$ for some prime $p$, or there is a $k$ such that $q=nk+1$ is prime and $1^k,\dots,n^k$ are all distinct modulo $q$ (in which case we call $q$ a strong representative of order $n$). The colorings obtained by this process we call multiplicative. We also show that nonmultiplicative colorings exist for some values of $n$.
There is an $n$-satisfactory coloring of $\mathbb Z^+$ if and only if there is such a coloring of the set $K_n$ of $n$-smooth numbers. We identify all $n$-satisfactory colorings for $n\le 5$ and all multiplicative colorings for $n\le 8$, and show that there are as many nonmultiplicative colorings of $K_n$ as there are real numbers for $n=6$ and 8. We show that if $n$ admits a strong representative $q$ then the set of such $q$ has positive natural density in the set of all primes.
We show that the question of whether there is an $n$-satisfactory coloring is equivalent to a problem about tilings, and use this to give a geometric characterization of multiplicative colorings.
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TL;DR: A general theorem is provided that covers a large family of invariants for which $\mathcal{G}(D)$ or $M$ is extremal among trees with degree sequence $D$ with respect to various graph invariants.
Abstract: The greedy tree $\mathcal{G}(D)$ and the $\mathcal{M}$-tree $\mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of invariants for which $\mathcal{G}(D)$ or $\mathcal{M}(D)$ is extremal. Many known results, for example on the Wiener index, the number of subtrees, the number of independent subsets and the number of matchings follow as corollaries, as do some new results on invariants such as the number of rooted spanning forests, the incidence energy and the solvability. We also extend our results on trees with fixed degree sequence $D$ to the set of trees whose degree sequence is majorised by a given sequence $D$, which also has a number of applications.
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TL;DR: All convex polyomino whose coordinate rings are Gorenstein are classified and the Castelnuovo-Mumford regularity of the coordinate ring of any stackpolyomino is computed in terms of the smallest interval which contains its vertices.
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TL;DR: The characterisation of non-separating planar graphs is used to prove that there are maximal linkless graphs with $3n-3$ edges which provides an answer to a question asked by Horst Sachs about the number of edges of linkless graph in 1983.
Abstract: A graph $G$ is a non-separating planar graph if there is a drawing $D$ of $G$ on the plane such that (1) no two edges cross each other in $D$ and (2) for any cycle $C$ in $D$, any two vertices not in $C$ are on the same side of $C$ in $D$.
Non-separating planar graphs are closed under taking minors and are a subclass of planar graphs and a superclass of outerplanar graphs.
In this paper, we show that a graph is a non-separating planar graph if and only if it does not contain $K_1 \cup K_4$ or $K_1 \cup K_{2,3}$ or $K_{1,1,3}$ as a minor.
Furthermore, we provide a structural characterisation of this class of graphs. More specifically, we show that any maximal non-separating planar graph is either an outerplanar graph or a subgraph of a wheel or it can be obtained by subdividing some of the side-edges of the 1-skeleton of a triangular prism (two disjoint triangles linked by a perfect matching).
Lastly, to demonstrate an application of non-separating planar graphs, we use the characterisation of non-separating planar graphs to prove that there are maximal linkless graphs with $3n-3$ edges which provides an answer to a question asked by Horst Sachs about the number of edges of linkless graphs in 1983.
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TL;DR: Three combinatorial models for symmetrized poly- Bernoulli numbers are introduced and generalizations of some identities for poly-BernoulliNumbers are derived.
Abstract: In this paper we introduce three combinatorial models for symmetrized poly-Bernoulli numbers. Based on our models we derive generalizations of some identities for poly-Bernoulli numbers. Finally, we set open questions and directions of further studies.
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TL;DR: It is shown that if $G$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi+G$ in the diameter $3$ case and to any group with every maximal subgroup normal.
Abstract: Funding: UK ESPRC grant number EP/R014604/1, and partially supported by a grant from the Simons Foundation (PJC, CMR-D); ESPRC grant number EP/R014604/1, St Leonard’s International Doctoral Fees Scholarship, School of Mathematics & Statistics PhD Funding Scholarship at the University of St Andrews (SDF).
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TL;DR: Borders are given on the maximum size of the families with ground set of the Bollobás set pairs if and only if there are at least $t$ distinct indices $i_1,i_2,\dots, i_k$.
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TL;DR: The method improved on Bukh--Jiang's method used in their 2017 publication reduced the best known upper bound by a factor of $\sqrt{5\log k}$.
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TL;DR: The authors' symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C.
Abstract: We compute, mimicking the Lascoux-Schutzenberger type A combinatorial procedure, left and right keys for a Kashiwara-Nakashima tableau in type C. These symplectic keys have a similar role as the keys for semistandard Young tableaux. More precisely, our symplectic keys give a tableau criterion for the Bruhat order on the hyperoctahedral group and cosets, and describe Demazure atoms and characters in type C. The right and the left symplectic keys are related through the Lusztig involution. A type C Schutzenberger evacuation is defined to realize that involution.
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TL;DR: It is shown that the symplectic-orthogonal supergroup $SpO(2|1)$ admits a cluster superalgebra structure and it is deduce that the supercommutative superal algebra generated by all the entries of a superfrieze is a subalgebra of a clustersuperalgebra.
Abstract: In this paper we propose the notion of cluster superalgebras which is a supersymmetric version of the classical cluster algebras introduced by Fomin and Zelevinsky. We show that the symplectic-orthogonal supergroup $SpO(2|1)$ admits a cluster superalgebra structure and as a consequence of this, we deduce that the supercommutative superalgebra generated by all the entries of a superfrieze is a subalgebra of a cluster superalgebra. We also show that the coordinate superalgebra of the super Grassmannian $G(2|0; 4|1)$ of chiral conformal superspace (that is, $(2|0)$ planes inside the superspace $\mathbb C^{4|1}$) is a quotient of a cluster superalgebra.
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TL;DR: The formulas that are avoided by every $\alpha$-free word for some $\alpha>1$ are characterized and progress is made toward the conjecture that every avoidable palindrome pattern is $4$-avoidable.
Abstract: We characterize the formulas that are avoided by every $\\alpha$-free word for some $\\alpha>1$. We show that the avoidable formulas whose fragments are of the form $XY$ or $XYX$ are $4$-avoidable. The largest avoidability index of an avoidable palindrome pattern is known to be at least $4$ and at most $16$. We make progress toward the conjecture that every avoidable palindrome pattern is $4$-avoidable.
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TL;DR: Bukh and Rudenko as mentioned in this paper showed that almost all permutations contain twins of length at most O(n 2/3 )/log n 1/3, where n is the length of a pair of disjoint order-isomorphic subsequences.
Abstract: Let $\pi$ be a permutation of the set $[n]=\{1,2,\dots, n\}$. Two disjoint order-isomorphic subsequences of $\pi$ are called twins. How long twins are contained in every permutation? The well known Erdős-Szekeres theorem implies that there is always a pair of twins of length $\Omega(\sqrt{n})$. On the other hand, by a simple probabilistic argument Gawron proved that for every $n\geqslant 1$ there exist permutations with all twins having length $O(n^{2/3})$. He conjectured that the latter bound is the correct size of the longest twins guaranteed in every permutation. We support this conjecture by showing that almost all permutations contain twins of length $\Omega(n^{2/3}/\log n^{1/3})$. Recently, Bukh and Rudenko have tweaked our proof and removed the log-factor. For completeness, we also present our version of their proof (see Remark 1.2 below on the interrelation between the two proofs).
In addition, we study several variants of the problem with diverse restrictions imposed on the twins. For instance, if we restrict attention to twins avoiding a fixed permutation $\tau$, then the corresponding extremal function equals $\Theta(\sqrt{n})$, provided that $\tau$ is not monotone. In case of block twins (each twin occupies a segment) we prove that it is $(1+o(1))\frac{\log n}{\log\log n}$, while for random permutations it is twice as large. For twins that jointly occupy a segment (tight twins), we prove that for every $n$ there are permutations avoiding them on all segments of length greater than $24$.
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TL;DR: This paper shows that the maximum possible value of $\varepsilon_3$ is $\frac16$ and disproves a conjecture of Gyarfas and Sarkozy.
Abstract: For every $n\in\mathbb{N}$ and $k\geq2$, it is known that every $k$-edge-colouring of the complete graph on $n$ vertices contains a monochromatic connected component of order at least $\frac{n}{k-1}$. For $k\geq3$, it is known that the complete graph can be replaced by a graph $G$ with $\delta(G)\geq(1-\varepsilon_k)n$ for some constant $\varepsilon_k$. In this paper, we show that the maximum possible value of $\varepsilon_3$ is $\frac16$. This disproves a conjecture of Gy\'{a}rfas and S\'{a}rk\"{o}zy.
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TL;DR: In this article, the permutree fans are realized by a permovahedron constructed from any realization of the braid fan, and the permutahedron can be realized by any permutrees fan.
Abstract: The associahedron is classically constructed as a removahedron, i.e. by deleting inequalities in the facet description of the permutahedron. This removahedral construction extends to all permutreehedra (which interpolate between the permutahedron, the associahedron and the cube). Here, we investigate removahedra constructions for all quotientopes (which realize the lattice quotients of the weak order). On the one hand, we observe that the permutree fans are the only quotient fans realized by a removahedron. On the other hand, we show that any permutree fan can be realized by a removahedron constructed from any realization of the braid fan. Our results finally lead to a complete description of the type cone of the permutree fans.
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TL;DR: Fulek and Keszegh proved that for every 0-1 matrix $P, either $sat(n, P) = O(1)$ or $sat (n,P) = \Theta(n)$, and affirm their conjecture by proving that almost all permutation matrices $P$ have “almost all” $k \times k$ permutations matrices such thatsat( n, P).
Abstract: Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that 0-1 matrix $A$ is saturating for the forbidden 0-1 matrix $P$ if $A$ avoids $P$ but changing any zero to a one in $A$ creates a copy of $P$. Define $sat(n, P)$ to be the minimum possible number of ones in an $n \times n$ 0-1 matrix that is saturating for $P$. Fulek and Keszegh proved that for every 0-1 matrix $P$, either $sat(n, P) = O(1)$ or $sat(n, P) = \Theta(n)$. They found two 0-1 matrices $P$ for which $sat(n, P) = O(1)$, as well as infinite families of 0-1 matrices $P$ for which $sat(n, P) = \Theta(n)$. Their results imply that $sat(n, P) = \Theta(n)$ for almost all $k \times k$ 0-1 matrices $P$.
Fulek and Keszegh conjectured that there are many more 0-1 matrices $P$ such that $sat(n, P) = O(1)$ besides the ones they found, and they asked for a characterization of all permutation matrices $P$ such that $sat(n, P) = O(1)$. We affirm their conjecture by proving that almost all $k \times k$ permutation matrices $P$ have $sat(n, P) = O(1)$. We also make progress on the characterization problem, since our proof of the main result exhibits a family of permutation matrices with bounded saturation functions.
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TL;DR: The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane.
Abstract: The main purpose of this survey is to provide an introduction, algebro-topological in nature, to Hirzebuch-type inequalities for plane curve arrangements in the complex projective plane. These inequalities gain more and more interest due to their utility in many combinatorial problems related to point or line arrangements in the plane. We would like to present a summary of the technicalities and also some recent applications, for instance in the context of the Weak Dirac Conjecture. We also advertise some open problems and questions.