Paul Bastide, Marthe Bonamy, Anthony Bonato, Pierre Charbit, Shahin Kamali, Théo Pierron, Mikaël Rabie
The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) le lceil sqrt{n}, rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the order of the graph is much larger than its growth, which is the maximal distance of a vertex to a well-chosen path in the graph. We prove that the conjecture for graphs of bounded growth reduces to a finite number of cases. We provide the best-known bound on the burning number of a connected graph $G$ of order $n,$ given by $b(G) le sqrt{4n/3} + 1,$ improving on the previously known $sqrt{3n/2}+O(1)$ bound. Using the improved upper bound, we show that the conjecture almost holds for all graphs with minimum degree at least $3$ and holds for all large enough graphs with minimum degree at least $4$. The previous best-known result was for graphs with minimum degree $23$.
燃烧数猜想认为，对于每一个阶为$n,$的连通图$G$，其燃烧数满足$b(G) le lceil sqrt{n}, rceil.$。在猜想保持开放状态的情况下，我们证明了当图的阶数远大于图的增长量(即图中一个顶点到一条选择路径的最大距离)时，它是渐近成立的。证明了有界增长图的猜想可以简化为有限种情况。我们通过$b(G) le sqrt{4n/3} + 1,$改进了之前已知的$sqrt{3n/2}+O(1)$界，给出了阶为$n,$的连通图$G$的燃烧数的最有名的界。利用改进的上界，我们证明了这个猜想几乎对所有最小度至少为$3$的图成立，对所有最小度至少为$4$的足够大的图成立。之前最著名的结果是最小度为$23$的图。
{"title":"Improved Pyrotechnics: Closer to the Burning Number Conjecture","authors":"Paul Bastide, Marthe Bonamy, Anthony Bonato, Pierre Charbit, Shahin Kamali, Théo Pierron, Mikaël Rabie","doi":"10.37236/11113","DOIUrl":"https://doi.org/10.37236/11113","url":null,"abstract":"The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) le lceil sqrt{n}, rceil.$ While the conjecture remains open, we prove that it is asymptotically true when the order of the graph is much larger than its growth, which is the maximal distance of a vertex to a well-chosen path in the graph. We prove that the conjecture for graphs of bounded growth reduces to a finite number of cases. We provide the best-known bound on the burning number of a connected graph $G$ of order $n,$ given by $b(G) le sqrt{4n/3} + 1,$ improving on the previously known $sqrt{3n/2}+O(1)$ bound. Using the improved upper bound, we show that the conjecture almost holds for all graphs with minimum degree at least $3$ and holds for all large enough graphs with minimum degree at least $4$. The previous best-known result was for graphs with minimum degree $23$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350960","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For integers $r geq 3$ and $t geq 2$, an $r$-uniform {em $t$-daisy} $D^t_r$ is a family of $binom{2t}{t}$ $r$-element sets of the form$${S cup T : Tsubset U, |T|=t }$$for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S cap U = emptyset$. It was conjectured by Bollobás, Leader and Malvenuto (and independently by Bukh) that the Turán densities of $t$-daisies satisfy $limlimits_{r to infty} pi(D_r^t) = 0$ for all $t geq 2$; this has become a well-known problem, and it is still open for all values of $t$. In this paper, we give lower bounds for the Turán densities of $r$-uniform $t$-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers $m geq 2t geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $mathbb{Z}/mmathbb{Z}$ such that for any $x in mathbb{Z}/mmathbb{Z}$ and any $2t$-element subset $X$ of $mathbb{Z}/mmathbb{Z}$, there are $t$ distinct elements of $X$ whose sum is not in the translate $x+R$? This is a slice-analogue of an extremal Hilbert cube problem considered by Gunderson and Rődl as well as Cilleruelo and Tesoro.
{"title":"Lower Bounds for the Turán Densities of Daisies","authors":"David Ellis, Dylan King","doi":"10.37236/11206","DOIUrl":"https://doi.org/10.37236/11206","url":null,"abstract":"For integers $r geq 3$ and $t geq 2$, an $r$-uniform {em $t$-daisy} $D^t_r$ is a family of $binom{2t}{t}$ $r$-element sets of the form$${S cup T : Tsubset U, |T|=t }$$for some sets $S,U$ with $|S|=r-t$, $|U|=2t$ and $S cap U = emptyset$. It was conjectured by Bollobás, Leader and Malvenuto (and independently by Bukh) that the Turán densities of $t$-daisies satisfy $limlimits_{r to infty} pi(D_r^t) = 0$ for all $t geq 2$; this has become a well-known problem, and it is still open for all values of $t$. In this paper, we give lower bounds for the Turán densities of $r$-uniform $t$-daisies. To do so, we introduce (and make some progress on) the following natural problem in additive combinatorics: for integers $m geq 2t geq 4$, what is the maximum cardinality $g(m,t)$ of a subset $R$ of $mathbb{Z}/mmathbb{Z}$ such that for any $x in mathbb{Z}/mmathbb{Z}$ and any $2t$-element subset $X$ of $mathbb{Z}/mmathbb{Z}$, there are $t$ distinct elements of $X$ whose sum is not in the translate $x+R$? This is a slice-analogue of an extremal Hilbert cube problem considered by Gunderson and Rődl as well as Cilleruelo and Tesoro.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.
{"title":"Cut Vertices in Random Planar Maps","authors":"Michael Drmota, Marc Noy, Benedikt Stufler","doi":"10.37236/11163","DOIUrl":"https://doi.org/10.37236/11163","url":null,"abstract":"The main goal of this paper is to determine the asymptotic behavior of the number $X_n$ of cut-vertices in random planar maps with $n$ edges. It is shown that $X_n/n to c$ in probability (for some explicit $c>0$). For so-called subcritical classes of planar maps (like outerplanar maps) we obtain a central limit theorem, too. Interestingly the combinatorics behind this seemingly simple problem is quite involved.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"74 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136010804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $G$ be a graph and let $alpha$ be a real number in $[0,1].$ In 2017, Nikiforov proposed the $A_alpha$-matrix for $G$ as $A_{alpha}(G)=alpha D(G)+(1-alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The largest eigenvalue of $A_{alpha}(G)$ is called the $A_alpha$-index of $G.$ The famous Erdős-Sós conjecture states that every $n$-vertex graph with more than $frac{1}{2}(k-1)n$ edges must contain every tree on $k+1$ vertices. In this paper, we consider an $A_alpha$-spectral version of this conjecture. For $n>k,$ let $S_{n,k}$ be the join of a clique on $k$ vertices with an independent set of $n-k$ vertices and denote by $S^+_{n,k}$ the graph obtained from $S_{n,k}$ by adding one edge. We show that for fixed $kgeq2,,0
{"title":"An Aα-Spectral Erdős-Sós Theorem","authors":"Ming-Zhu Chen, Shuchao Li, Zhao-Ming Li, Yuantian Yu, Xiao-Dong Zhang","doi":"10.37236/11593","DOIUrl":"https://doi.org/10.37236/11593","url":null,"abstract":"Let $G$ be a graph and let $alpha$ be a real number in $[0,1].$ In 2017, Nikiforov proposed the $A_alpha$-matrix for $G$ as $A_{alpha}(G)=alpha D(G)+(1-alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The largest eigenvalue of $A_{alpha}(G)$ is called the $A_alpha$-index of $G.$ The famous Erdős-Sós conjecture states that every $n$-vertex graph with more than $frac{1}{2}(k-1)n$ edges must contain every tree on $k+1$ vertices. In this paper, we consider an $A_alpha$-spectral version of this conjecture. For $n>k,$ let $S_{n,k}$ be the join of a clique on $k$ vertices with an independent set of $n-k$ vertices and denote by $S^+_{n,k}$ the graph obtained from $S_{n,k}$ by adding one edge. We show that for fixed $kgeq2,,0<alpha<1$ and $ngeqfrac{88k^2(k+1)^2}{alpha^4(1-alpha)}$, if a graph on $n$ vertices has $A_alpha$-index at least as large as $S_{n,k}$ (resp. $S^+_{n,k}$), then it contains all trees on $2k+2$ (resp. $2k+3$) vertices, or it is isomorphic to $S_{n,k}$ (resp. $S^+_{n,k}$). These extend the results of Cioabă, Desai and Tait (2022), in which they confirmed the adjacency spectral version of the Erdős-Sós conjecture.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136011694","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Linda Cook, Tomáš Masařík, Marcin Pilipczuk, Amadeus Reinald, Uéverton S. Souza
An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $omega(D)$ has chromatic number at most $f(omega(D))$. Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs, Electron. J. Comb., 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic.
Aboulker, Charbit, and Naserasr’s $overrightarrow{chi}$ -boundedness conjecture states that for every oriented forest $F$, there is some function f such that every $F$-free oriented graph $D$ has dichromatic number at most $f(omega(D))$, where $omega(D)$ is the size of a maximum clique in the graph underlying $D$. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr’s $overrightarrow{chi}$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.
有向图是不包含长度为2的有向环的有向图。如果$D$不包含$H$作为诱导子(di)图，则(定向)图$D$与$H$无关。Gyárfás-Sumner猜想是一个关于简单图的广开猜想，它表明对于任何森林$F$，存在一个函数$f$，使得每一个团数$omega(D)$的$F$自由图$G$最多有一个色数$f(omega(D))$。Aboulker, Charbit, and Naserasr [Gyárfás-Sumner猜想对有向图的扩展，电子。]J. Comb。[j][2021]提出了这个猜想对有向图的二色数的类比。有向图$D$的二色数是为$D$的顶点集上色所需的最小颜色数，这样$D$中的有向循环就不会是单色的。＆＃x0D;Aboulker, Charbit和Naserasr的$overrightarrow{chi}$ -有界猜想指出，对于每个有向森林$F$，存在一些函数f，使得每个$F$自由有向图$D$最多有二色数$f(omega(D))$，其中$omega(D)$是$D$下面的图中最大团的大小。在本文中，我们通过证明当$F$是四个顶点上的路径的任意方向时，它成立，执行了证明Aboulker, Charbit和Naserasr的$overrightarrow{chi}$有界猜想的第一步。
{"title":"Proving a Directed Analogue of the Gyárfás-Sumner Conjecture for Orientations of $P_4$","authors":"Linda Cook, Tomáš Masařík, Marcin Pilipczuk, Amadeus Reinald, Uéverton S. Souza","doi":"10.37236/11538","DOIUrl":"https://doi.org/10.37236/11538","url":null,"abstract":"An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $omega(D)$ has chromatic number at most $f(omega(D))$. Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs, Electron. J. Comb., 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic.
 Aboulker, Charbit, and Naserasr’s $overrightarrow{chi}$ -boundedness conjecture states that for every oriented forest $F$, there is some function f such that every $F$-free oriented graph $D$ has dichromatic number at most $f(omega(D))$, where $omega(D)$ is the size of a maximum clique in the graph underlying $D$. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr’s $overrightarrow{chi}$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"72 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136011693","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jesper Nederlof, Michał Pilipczuk, Karol Węgrzycki
We study Koebe orderings of planar graphs: vertex orderings obtained by modelling the graph as the intersection graph of pairwise internally-disjoint discs in the plane, and ordering the vertices by non-increasing radii of the associated discs. We prove that for every $din mathbb{N}$, any such ordering has $d$-admissibility bounded by $O(d/ln d)$ and weak $d$-coloring number bounded by $O(d^4 ln d)$. This in particular shows that the $d$-admissibility of planar graphs is bounded by $O(d/ln d)$, which asymptotically matches a known lower bound due to Dvořák and Siebertz.
{"title":"Bounding Generalized Coloring Numbers of Planar Graphs Using Coin Models","authors":"Jesper Nederlof, Michał Pilipczuk, Karol Węgrzycki","doi":"10.37236/11095","DOIUrl":"https://doi.org/10.37236/11095","url":null,"abstract":"We study Koebe orderings of planar graphs: vertex orderings obtained by modelling the graph as the intersection graph of pairwise internally-disjoint discs in the plane, and ordering the vertices by non-increasing radii of the associated discs. We prove that for every $din mathbb{N}$, any such ordering has $d$-admissibility bounded by $O(d/ln d)$ and weak $d$-coloring number bounded by $O(d^4 ln d)$. This in particular shows that the $d$-admissibility of planar graphs is bounded by $O(d/ln d)$, which asymptotically matches a known lower bound due to Dvořák and Siebertz.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136011691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A signed graph $(G, sigma)$ is a graph $G$ together with an assignment $sigma:E(G) rightarrow {+,-}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between the theories of minors and colorings of graphs. Following this thread of thoughts, we investigate this connection through the notion of Extended Double Covers of signed graphs, which was recently introduced by Naserasr, Sopena and Zaslavsky. More precisely, we say that a signed graph $(B, pi)$ is planar-complete if any signed planar graph $(G, sigma)$ which verifies the conditions of a basic no-homomorphism lemma with respect to $(B,pi)$ admits a homomorphism to $(B, pi)$. Our conjecture then is that: if $(B, pi)$ is a connected signed graph with no positive odd closed walk which is planar-complete, then its Extended Double Cover ${rm EDC}(B,pi)$ is also planar-complete. We observe that this conjecture largely extends the Four-Color Theorem and is strongly connected to a number of conjectures in extension of this famous theorem. A given (signed) graph $(B,pi)$ bounds a class of (signed) graphs if every (signed) graph in the class admits a homomorphism to $(B,pi)$.In this work, and in support of our conjecture, we prove it for the subclass of signed $K_4$-minor free graphs. Inspired by this development, we then investigate the problem of finding optimal homomorphism bounds for subclasses of signed $K_4$-minor-free graphs with restrictions on their girth and we present nearly optimal solutions. Our work furthermore leads to the development of weighted signed graphs.
{"title":"Extended Double Covers and Homomorphism Bounds of Signed Graphs","authors":"Florent Foucaud, Reza Naserasr, Rongxing Xu","doi":"10.37236/10754","DOIUrl":"https://doi.org/10.37236/10754","url":null,"abstract":"A signed graph $(G, sigma)$ is a graph $G$ together with an assignment $sigma:E(G) rightarrow {+,-}$. The notion of homomorphisms of signed graphs is a relatively new development which allows to strengthen the connection between the theories of minors and colorings of graphs. Following this thread of thoughts, we investigate this connection through the notion of Extended Double Covers of signed graphs, which was recently introduced by Naserasr, Sopena and Zaslavsky. More precisely, we say that a signed graph $(B, pi)$ is planar-complete if any signed planar graph $(G, sigma)$ which verifies the conditions of a basic no-homomorphism lemma with respect to $(B,pi)$ admits a homomorphism to $(B, pi)$. Our conjecture then is that: if $(B, pi)$ is a connected signed graph with no positive odd closed walk which is planar-complete, then its Extended Double Cover ${rm EDC}(B,pi)$ is also planar-complete. We observe that this conjecture largely extends the Four-Color Theorem and is strongly connected to a number of conjectures in extension of this famous theorem. A given (signed) graph $(B,pi)$ bounds a class of (signed) graphs if every (signed) graph in the class admits a homomorphism to $(B,pi)$.In this work, and in support of our conjecture, we prove it for the subclass of signed $K_4$-minor free graphs. Inspired by this development, we then investigate the problem of finding optimal homomorphism bounds for subclasses of signed $K_4$-minor-free graphs with restrictions on their girth and we present nearly optimal solutions. Our work furthermore leads to the development of weighted signed graphs.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136010813","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.
{"title":"Quantum Continuants, Quantum Rotundus and Triangulations of Annuli","authors":"Ludivine Leclere, Sophie Morier-Genoud","doi":"10.37236/11400","DOIUrl":"https://doi.org/10.37236/11400","url":null,"abstract":"We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136011692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yuval Filmus, Edward A. Hirsch, Sascha Kurz, Ferdinand Ihringer, Artur Ryazanov, Alexander V. Smal, Marc Vinyals
A subcube partition is a partition of the Boolean cube ${0,1}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates.
We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of ${0,dots,q-1}^n$, and partitions of $mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size.
Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.
{"title":"Irreducible Subcube Partitions","authors":"Yuval Filmus, Edward A. Hirsch, Sascha Kurz, Ferdinand Ihringer, Artur Ryazanov, Alexander V. Smal, Marc Vinyals","doi":"10.37236/11862","DOIUrl":"https://doi.org/10.37236/11862","url":null,"abstract":"A subcube partition is a partition of the Boolean cube ${0,1}^n$ into subcubes. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. A subcube partition is tight if it “mentions” all coordinates.
 We study extremal properties of tight irreducible subcube partitions: minimal size, minimal weight, maximal number of points, maximal size, and maximal minimum dimension. We also consider the existence of homogeneous tight irreducible subcube partitions, in which all subcubes have the same dimensions. We additionally study subcube partitions of ${0,dots,q-1}^n$, and partitions of $mathbb{F}_2^n$ into affine subspaces, in both cases focusing on the minimal size.
 Our constructions and computer experiments lead to several conjectures on the extremal values of the aforementioned properties.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136298345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Yule branching process is a classical model for the random generation of gene tree topologies in population genetics. It generates binary ranked trees -also called histories- with a finite number $n$ of leaves. We study the lengths $ell_1 > ell_2 > cdots > ell_k > cdots$ of the external branches of a Yule generated random history of size $n$, where the length of an external branch is defined as the rank of its parent node. When $n rightarrow infty$, we show that the random variable $ell_k$, once rescaled as $frac{n-ell_k}{sqrt{n/2}}$, follows a $chi$-distribution with $2k$ degrees of freedom, with mean $mathbb E(ell_k) sim n$ and variance $mathbb V(ell_k) sim n big(k-frac{pi k^2}{16^k} binom{2k}{k}^2big)$. Our results contribute to the study of the combinatorial features of Yule generated gene trees, in which external branches are associated with singleton mutations affecting individual gene copies.
{"title":"Distribution of External Branch Lengths in Yule Histories","authors":"F. Disanto, Michael Fuchs","doi":"10.37236/11438","DOIUrl":"https://doi.org/10.37236/11438","url":null,"abstract":"The Yule branching process is a classical model for the random generation of gene tree topologies in population genetics. It generates binary ranked trees -also called histories- with a finite number $n$ of leaves. We study the lengths $ell_1 > ell_2 > cdots > ell_k > cdots$ of the external branches of a Yule generated random history of size $n$, where the length of an external branch is defined as the rank of its parent node. When $n rightarrow infty$, we show that the random variable $ell_k$, once rescaled as $frac{n-ell_k}{sqrt{n/2}}$, follows a $chi$-distribution with $2k$ degrees of freedom, with mean $mathbb E(ell_k) sim n$ and variance $mathbb V(ell_k) sim n big(k-frac{pi k^2}{16^k} binom{2k}{k}^2big)$. Our results contribute to the study of the combinatorial features of Yule generated gene trees, in which external branches are associated with singleton mutations affecting individual gene copies.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"224 ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72441867","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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