In this thesis, we present quantitative Ramsey-type results in the setting of finite sets that are equipped with a partial order, so-called posets. A prominent example of a poset is the Boolean lattice $Q_n$, which consists of all subsets of ${1,dots,n}$, ordered by inclusion. For posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest $N$ such that no matter how the elements of $Q_N$ are colored in blue and red, there is either an induced subposet isomorphic to $P$ in which every element is colored blue, or an induced subposet isomorphic to $Q$ in which every element is colored red. The central focus of this thesis is to investigate $R(P,Q_n)$, where $P$ is fixed and $n$ grows large. Our results contribute to an active area of discrete mathematics, which studies the existence of large homogeneous substructures in host structures with local constraints, introduced for graphs by ErdH{o}s and Hajnal. We provide an asymptotically tight bound on $R(P,Q_n)$ for $P$ from several classes of posets, and show a dichotomy in the asymptotic behavior of $R(P,Q_n)$, depending on whether $P$ contains a subposet isomorphic to one of two specific posets. A fundamental question in the study of poset Ramsey numbers is to determine the asymptotic behavior of $R(Q_n,Q_n)$ for large $n$. In this dissertation, we present improvements on the known lower and upper bound on $R(Q_n,Q_n)$. Moreover, we explore variations of the poset Ramsey setting, including ErdH{o}s-Hajnal-type questions when the small forbidden poset has a non-monochromatic color pattern, and so-called weak poset Ramsey numbers, which are concerned with non-induced subposets.
{"title":"Ramsey numbers for partially ordered sets","authors":"Christian Winter","doi":"arxiv-2409.08819","DOIUrl":"https://doi.org/arxiv-2409.08819","url":null,"abstract":"In this thesis, we present quantitative Ramsey-type results in the setting of\u0000finite sets that are equipped with a partial order, so-called posets. A\u0000prominent example of a poset is the Boolean lattice $Q_n$, which consists of\u0000all subsets of ${1,dots,n}$, ordered by inclusion. For posets $P$ and $Q$,\u0000the poset Ramsey number $R(P,Q)$ is the smallest $N$ such that no matter how\u0000the elements of $Q_N$ are colored in blue and red, there is either an induced\u0000subposet isomorphic to $P$ in which every element is colored blue, or an\u0000induced subposet isomorphic to $Q$ in which every element is colored red. The central focus of this thesis is to investigate $R(P,Q_n)$, where $P$ is\u0000fixed and $n$ grows large. Our results contribute to an active area of discrete\u0000mathematics, which studies the existence of large homogeneous substructures in\u0000host structures with local constraints, introduced for graphs by ErdH{o}s and\u0000Hajnal. We provide an asymptotically tight bound on $R(P,Q_n)$ for $P$ from\u0000several classes of posets, and show a dichotomy in the asymptotic behavior of\u0000$R(P,Q_n)$, depending on whether $P$ contains a subposet isomorphic to one of\u0000two specific posets. A fundamental question in the study of poset Ramsey numbers is to determine\u0000the asymptotic behavior of $R(Q_n,Q_n)$ for large $n$. In this dissertation, we\u0000present improvements on the known lower and upper bound on $R(Q_n,Q_n)$.\u0000Moreover, we explore variations of the poset Ramsey setting, including\u0000ErdH{o}s-Hajnal-type questions when the small forbidden poset has a\u0000non-monochromatic color pattern, and so-called weak poset Ramsey numbers, which\u0000are concerned with non-induced subposets.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Boštjan Brešar, Sandi Klavžar, Babak Samadi, Ismael G. Yero
Two relationships between the injective chromatic number and, respectively, chromatic number and chromatic index, are proved. They are applied to determine the injective chromatic number of Sierpi'nski graphs and to give a short proof that Sierpi'nski graphs are Class $1$. Sierpi'nski-like graphs are also considered, including generalized Sierpi'nski graphs over cycles and rooted products. It is proved that the injective chromatic number of a rooted product of two graphs lies in a set of six possible values. Sierpi'nski graphs and Kneser graphs $K(n,r)$ are considered with respect of being perfect injectively colorable, where a graph is perfect injectively colorable if it has an injective coloring in which every color class forms an open packing of largest cardinality. In particular, all Sierpi'nski graphs and Kneser graphs $K(n, r)$ with $n ge 3r-1$ are perfect injectively colorable graph, while $K(7,3)$ is not.
{"title":"Injective colorings of Sierpiński-like graphs and Kneser graphs","authors":"Boštjan Brešar, Sandi Klavžar, Babak Samadi, Ismael G. Yero","doi":"arxiv-2409.08856","DOIUrl":"https://doi.org/arxiv-2409.08856","url":null,"abstract":"Two relationships between the injective chromatic number and, respectively,\u0000chromatic number and chromatic index, are proved. They are applied to determine\u0000the injective chromatic number of Sierpi'nski graphs and to give a short proof\u0000that Sierpi'nski graphs are Class $1$. Sierpi'nski-like graphs are also\u0000considered, including generalized Sierpi'nski graphs over cycles and rooted\u0000products. It is proved that the injective chromatic number of a rooted product\u0000of two graphs lies in a set of six possible values. Sierpi'nski graphs and\u0000Kneser graphs $K(n,r)$ are considered with respect of being perfect injectively\u0000colorable, where a graph is perfect injectively colorable if it has an\u0000injective coloring in which every color class forms an open packing of largest\u0000cardinality. In particular, all Sierpi'nski graphs and Kneser graphs $K(n, r)$\u0000with $n ge 3r-1$ are perfect injectively colorable graph, while $K(7,3)$ is\u0000not.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathcal{H}$ be a graph class and $kinmathbb{N}$. We say a graph $G$ admits a emph{$k$-identification to $mathcal{H}$} if there is a partition $mathcal{P}$ of some set $Xsubseteq V(G)$ of size at most $k$ such that after identifying each part in $mathcal{P}$ to a single vertex, the resulting graph belongs to $mathcal{H}$. The graph parameter ${sf id}_{mathcal{H}}$ is defined so that ${sf id}_{mathcal{H}}(G)$ is the minimum $k$ such that $G$ admits a $k$-identification to $mathcal{H}$, and the problem of textsc{Identification to $mathcal{H}$} asks, given a graph $G$ and $kinmathbb{N}$, whether ${sf id}_{mathcal{H}}(G)le k$. If we set $mathcal{H}$ to be the class $mathcal{F}$ of acyclic graphs, we generate the problem textsc{Identification to Forest}, which we show to be {sf NP}-complete. We prove that, when parameterized by the size $k$ of the identification set, it admits a kernel of size $2k+1$. For our kernel we reveal a close relation of textsc{Identification to Forest} with the textsc{Vertex Cover} problem. We also study the combinatorics of the textsf{yes}-instances of textsc{Identification to $mathcal{H}$}, i.e., the class $mathcal{H}^{(k)}:={Gmid {sf id}_{mathcal{H}}(G)le k}$, {which we show to be minor-closed for every $k$} when $mathcal{H}$ is minor-closed. We prove that the minor-obstructions of $mathcal{F}^{(k)}$ are of size at most $2k+4$. We also prove that every graph $G$ such that ${sf id}_{mathcal{F}}(G)$ is sufficiently big contains as a minor either a cycle on $k$ vertices, or $k$ disjoint triangles, or the emph{$k$-marguerite} graph, that is the graph obtained by $k$ disjoint triangles by identifying one vertex of each of them into the same vertex.
{"title":"Vertex identification to a forest","authors":"Laure Morelle, Ignasi Sau, Dimitrios M. Thilikos","doi":"arxiv-2409.08883","DOIUrl":"https://doi.org/arxiv-2409.08883","url":null,"abstract":"Let $mathcal{H}$ be a graph class and $kinmathbb{N}$. We say a graph $G$\u0000admits a emph{$k$-identification to $mathcal{H}$} if there is a partition\u0000$mathcal{P}$ of some set $Xsubseteq V(G)$ of size at most $k$ such that after\u0000identifying each part in $mathcal{P}$ to a single vertex, the resulting graph\u0000belongs to $mathcal{H}$. The graph parameter ${sf id}_{mathcal{H}}$ is\u0000defined so that ${sf id}_{mathcal{H}}(G)$ is the minimum $k$ such that $G$\u0000admits a $k$-identification to $mathcal{H}$, and the problem of\u0000textsc{Identification to $mathcal{H}$} asks, given a graph $G$ and\u0000$kinmathbb{N}$, whether ${sf id}_{mathcal{H}}(G)le k$. If we set\u0000$mathcal{H}$ to be the class $mathcal{F}$ of acyclic graphs, we generate the\u0000problem textsc{Identification to Forest}, which we show to be {sf\u0000NP}-complete. We prove that, when parameterized by the size $k$ of the\u0000identification set, it admits a kernel of size $2k+1$. For our kernel we reveal\u0000a close relation of textsc{Identification to Forest} with the textsc{Vertex\u0000Cover} problem. We also study the combinatorics of the textsf{yes}-instances\u0000of textsc{Identification to $mathcal{H}$}, i.e., the class\u0000$mathcal{H}^{(k)}:={Gmid {sf id}_{mathcal{H}}(G)le k}$, {which we show\u0000to be minor-closed for every $k$} when $mathcal{H}$ is minor-closed. We prove\u0000that the minor-obstructions of $mathcal{F}^{(k)}$ are of size at most $2k+4$.\u0000We also prove that every graph $G$ such that ${sf id}_{mathcal{F}}(G)$ is\u0000sufficiently big contains as a minor either a cycle on $k$ vertices, or $k$\u0000disjoint triangles, or the emph{$k$-marguerite} graph, that is the graph\u0000obtained by $k$ disjoint triangles by identifying one vertex of each of them\u0000into the same vertex.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The textit{minimum positive co-degree} of a non-empty $r$-graph $H$, denoted $delta_{r-1}^+(H)$, is the largest integer $k$ such that if a set $S subset V(H)$ of size $r-1$ is contained in at least one $r$-edge of $H$, then $S$ is contained in at least $k$ $r$-edges of $H$. Motivated by several recent papers which study minimum positive co-degree as a reasonable notion of minimum degree in $r$-graphs, we consider bounds of $delta_{r-1}^+(H)$ which will guarantee the existence of various spanning subgraphs in $H$. We precisely determine the minimum positive co-degree threshold for Berge Hamiltonian cycles in $r$-graphs, and asymptotically determine the minimum positive co-degree threshold for loose Hamiltonian cycles in $3$-graphs. For all $r$, we also determine up to an additive constant the minimum positive co-degree threshold for perfect matchings.
{"title":"Positive co-degree thresholds for spanning structures","authors":"Anastasia Halfpap, Van Magnan","doi":"arxiv-2409.09185","DOIUrl":"https://doi.org/arxiv-2409.09185","url":null,"abstract":"The textit{minimum positive co-degree} of a non-empty $r$-graph $H$, denoted\u0000$delta_{r-1}^+(H)$, is the largest integer $k$ such that if a set $S subset\u0000V(H)$ of size $r-1$ is contained in at least one $r$-edge of $H$, then $S$ is\u0000contained in at least $k$ $r$-edges of $H$. Motivated by several recent papers\u0000which study minimum positive co-degree as a reasonable notion of minimum degree\u0000in $r$-graphs, we consider bounds of $delta_{r-1}^+(H)$ which will guarantee\u0000the existence of various spanning subgraphs in $H$. We precisely determine the\u0000minimum positive co-degree threshold for Berge Hamiltonian cycles in\u0000$r$-graphs, and asymptotically determine the minimum positive co-degree\u0000threshold for loose Hamiltonian cycles in $3$-graphs. For all $r$, we also\u0000determine up to an additive constant the minimum positive co-degree threshold\u0000for perfect matchings.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142269005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
If $A$ is a finite Abelian group, then a labeling $f colon E (G) rightarrow A$ of the edges of some graph $G$ induces a vertex labeling on $G$; the vertex $u$ receives the label $sum_{vin N(u)}f (v)$, where $N(u)$ is an open neighborhood of the vertex $u$. A graph $G$ is $E_A$-cordial if there is an edge-labeling such that (1) the edge label classes differ in size by at most one and (2) the induced vertex label classes differ in size by at most one. Such a labeling is called $E_A$-cordial. In the literature, so far only $E_A$-cordial labeling in cyclic groups has been studied. The corresponding problem was studied by Kaplan, Lev and Roditty. Namely, they introduced $A^*$-antimagic labeling as a generalization of antimagic labeling cite{ref_KapLevRod}. Simply saying, for a tree of order $|A|$ the $A^*$-antimagic labeling is such $E_A$-cordial labeling that the label $0$ is prohibited on the edges. In this paper, we give necessary and sufficient conditions for paths to be $E_A$-cordial for any cyclic $A$. We also show that the conjecture for $A^*$-antimagic labeling of trees posted in cite{ref_KapLevRod} is not true.
{"title":"$E_A$-cordial labeling of graphs and its implications for $A$-antimagic labeling of trees","authors":"Sylwia Cichacz","doi":"arxiv-2409.09136","DOIUrl":"https://doi.org/arxiv-2409.09136","url":null,"abstract":"If $A$ is a finite Abelian group, then a labeling $f colon E (G) rightarrow\u0000A$ of the edges of some graph $G$ induces a vertex labeling on $G$; the vertex\u0000$u$ receives the label $sum_{vin N(u)}f (v)$, where $N(u)$ is an open\u0000neighborhood of the vertex $u$. A graph $G$ is $E_A$-cordial if there is an\u0000edge-labeling such that (1) the edge label classes differ in size by at most\u0000one and (2) the induced vertex label classes differ in size by at most one.\u0000Such a labeling is called $E_A$-cordial. In the literature, so far only\u0000$E_A$-cordial labeling in cyclic groups has been studied. The corresponding problem was studied by Kaplan, Lev and Roditty. Namely,\u0000they introduced $A^*$-antimagic labeling as a generalization of antimagic\u0000labeling cite{ref_KapLevRod}. Simply saying, for a tree of order $|A|$ the\u0000$A^*$-antimagic labeling is such $E_A$-cordial labeling that the label $0$ is\u0000prohibited on the edges. In this paper, we give necessary and sufficient conditions for paths to be\u0000$E_A$-cordial for any cyclic $A$. We also show that the conjecture for\u0000$A^*$-antimagic labeling of trees posted in cite{ref_KapLevRod} is not true.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study lattice polytopes which arise as the convex hull of chip vectors for textit{self-reachable} chip configurations on a tree $T$. We show that these polytopes always have the integer decomposition property and characterize the vertex sets of these polytopes. Additionally, in the case of self-reachable configurations with the smallest possible number of chips, we show that these polytopes are unimodularly equivalent to a unit cube.
{"title":"Self-Reachable Configuration Polytopes for Trees","authors":"Benjamin Lyons, McCabe Olsen","doi":"arxiv-2409.07675","DOIUrl":"https://doi.org/arxiv-2409.07675","url":null,"abstract":"We study lattice polytopes which arise as the convex hull of chip vectors for\u0000textit{self-reachable} chip configurations on a tree $T$. We show that these\u0000polytopes always have the integer decomposition property and characterize the\u0000vertex sets of these polytopes. Additionally, in the case of self-reachable\u0000configurations with the smallest possible number of chips, we show that these\u0000polytopes are unimodularly equivalent to a unit cube.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $U_{n,d}$ be the uniform matroid of rank $d$ on $n$ elements. Denote by $g_{U_{n,d}}(t)$ the Speyer's $g$-polynomial of $U_{n,d}$. The Tur'{a}n inequality and higher order Tur'{a}n inequality are related to the Laguerre-P'{o}lya ($mathcal{L}$-$mathcal{P}$) class of real entire functions, and the $mathcal{L}$-$mathcal{P}$ class has close relation with the Riemann hypothesis. The Tur'{a}n type inequalities have received much attention. Infinite log-concavity is also a deep generalization of Tur'{a}n inequality with different direction. In this paper, we mainly obtain the infinite log-concavity and the higher order Tur'{a}n inequality of the sequence ${g_{U_{n,d}}(t)}_{d=1}^{n-1}$ for $t>0$. In order to prove these results, we show that the generating function of $g_{U_{n,d}}(t)$, denoted $h_n(x;t)$, has only real zeros for $t>0$. Consequently, for $t>0$, we also obtain the $gamma$-positivity of the polynomial $h_n(x;t)$, the asymptotical normality of $g_{U_{n,d}}(t)$, and the Laguerre inequalities for $g_{U_{n,d}}(t)$ and $h_n(x;t)$.
{"title":"Infinite log-concavity and higher order Turán inequality for Speyer's $g$-polynomial of uniform matroids","authors":"James J. Y. Zhao","doi":"arxiv-2409.08085","DOIUrl":"https://doi.org/arxiv-2409.08085","url":null,"abstract":"Let $U_{n,d}$ be the uniform matroid of rank $d$ on $n$ elements. Denote by\u0000$g_{U_{n,d}}(t)$ the Speyer's $g$-polynomial of $U_{n,d}$. The Tur'{a}n\u0000inequality and higher order Tur'{a}n inequality are related to the\u0000Laguerre-P'{o}lya ($mathcal{L}$-$mathcal{P}$) class of real entire\u0000functions, and the $mathcal{L}$-$mathcal{P}$ class has close relation with\u0000the Riemann hypothesis. The Tur'{a}n type inequalities have received much\u0000attention. Infinite log-concavity is also a deep generalization of Tur'{a}n\u0000inequality with different direction. In this paper, we mainly obtain the\u0000infinite log-concavity and the higher order Tur'{a}n inequality of the\u0000sequence ${g_{U_{n,d}}(t)}_{d=1}^{n-1}$ for $t>0$. In order to prove these\u0000results, we show that the generating function of $g_{U_{n,d}}(t)$, denoted\u0000$h_n(x;t)$, has only real zeros for $t>0$. Consequently, for $t>0$, we also\u0000obtain the $gamma$-positivity of the polynomial $h_n(x;t)$, the asymptotical\u0000normality of $g_{U_{n,d}}(t)$, and the Laguerre inequalities for\u0000$g_{U_{n,d}}(t)$ and $h_n(x;t)$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tesshu Hanaka, Yuni Iwamasa, Yasuaki Kobayashi, Yuto Okada, Rin Saito
Given a graph $G$ and two spanning trees $T$ and $T'$ in $G$, Spanning Tree Reconfiguration asks whether there is a step-by-step transformation from $T$ to $T'$ such that all intermediates are also spanning trees of $G$, by exchanging an edge in $T$ with an edge outside $T$ at a single step. This problem is naturally related to matroid theory, which shows that there always exists such a transformation for any pair of $T$ and $T'$. Motivated by this example, we study the problem of transforming a sequence of spanning trees into another sequence of spanning trees. We formulate this problem in the language of matroid theory: Given two sequences of bases of matroids, the goal is to decide whether there is a transformation between these sequences. We design a polynomial-time algorithm for this problem, even if the matroids are given as basis oracles. To complement this algorithmic result, we show that the problem of finding a shortest transformation is NP-hard to approximate within a factor of $c log n$ for some constant $c > 0$, where $n$ is the total size of the ground sets of the input matroids.
{"title":"Basis sequence reconfiguration in the union of matroids","authors":"Tesshu Hanaka, Yuni Iwamasa, Yasuaki Kobayashi, Yuto Okada, Rin Saito","doi":"arxiv-2409.07848","DOIUrl":"https://doi.org/arxiv-2409.07848","url":null,"abstract":"Given a graph $G$ and two spanning trees $T$ and $T'$ in $G$, Spanning Tree\u0000Reconfiguration asks whether there is a step-by-step transformation from $T$ to\u0000$T'$ such that all intermediates are also spanning trees of $G$, by exchanging\u0000an edge in $T$ with an edge outside $T$ at a single step. This problem is\u0000naturally related to matroid theory, which shows that there always exists such\u0000a transformation for any pair of $T$ and $T'$. Motivated by this example, we\u0000study the problem of transforming a sequence of spanning trees into another\u0000sequence of spanning trees. We formulate this problem in the language of\u0000matroid theory: Given two sequences of bases of matroids, the goal is to decide\u0000whether there is a transformation between these sequences. We design a\u0000polynomial-time algorithm for this problem, even if the matroids are given as\u0000basis oracles. To complement this algorithmic result, we show that the problem\u0000of finding a shortest transformation is NP-hard to approximate within a factor\u0000of $c log n$ for some constant $c > 0$, where $n$ is the total size of the\u0000ground sets of the input matroids.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Driven by applications in the natural, social and computer sciences several algorithms have been proposed to enumerate all sets $X$ of vertices of a graph $G$ that induce a connected subgraph. Our algorithm AllMetricSets enumerates all $X$'s that induce (more exquisite) metric subgraphs. Here "metric" means that any distinct $s,tin X$ are joined by a globally shortest $s-t$ path.
{"title":"Compression with wildcards: All induced metric subgraphs","authors":"Marcel Wild","doi":"arxiv-2409.08363","DOIUrl":"https://doi.org/arxiv-2409.08363","url":null,"abstract":"Driven by applications in the natural, social and computer sciences several\u0000algorithms have been proposed to enumerate all sets $X$ of vertices of a graph\u0000$G$ that induce a connected subgraph. Our algorithm AllMetricSets enumerates\u0000all $X$'s that induce (more exquisite) metric subgraphs. Here \"metric\" means\u0000that any distinct $s,tin X$ are joined by a globally shortest $s-t$ path.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255028","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let $rho(d+1,k) = frac{1}{2}[{lceil (d+1)/2 rceil choose d-k} + {lfloor (d+1)/2 rfloor choose d-k}]$. Recently, the author, answering B'ar'any's question from 1998, proved that for all $lfloor frac{d-1}{2} rfloor leq k leq d$, [ f_k(mathscr{X}) geq rho(d+1,k)f_d(mathscr{X}). ] We prove a generalization: if $mathscr{X}$ is a shellable, strongly regular CW sphere or CW ball of dimension $d$, then for all $lfloor frac{d-1}{2} rfloor leq k leq d$, [ f_k(mathscr{X}) geq rho(d+1,k)f_d(mathscr{X}) + frac{1}{2}f_k(partial mathscr{X}), ] with equality precisely when $k=d$ or when $k=d-1$ and $mathscr{X}$ is simplicial. We further prove that if $mathscr{S}$ is a strongly regular CW sphere of dimension $d$, and the face poset of $mathscr{S}$ is both CL-shellable and dual CL-shellable, then $f_k(mathscr{S}) geq min{f_0(mathscr{S}),f_d(mathscr{S})}$ for all $0 leq k leq d$.
{"title":"Face Numbers of Shellable CW Balls and Spheres","authors":"Joshua Hinman","doi":"arxiv-2409.08427","DOIUrl":"https://doi.org/arxiv-2409.08427","url":null,"abstract":"Let $mathscr{X}$ be the boundary complex of a $(d+1)$-polytope, and let\u0000$rho(d+1,k) = frac{1}{2}[{lceil (d+1)/2 rceil choose d-k} + {lfloor\u0000(d+1)/2 rfloor choose d-k}]$. Recently, the author, answering B'ar'any's\u0000question from 1998, proved that for all $lfloor frac{d-1}{2} rfloor leq k\u0000leq d$, [ f_k(mathscr{X}) geq rho(d+1,k)f_d(mathscr{X}). ] We prove a\u0000generalization: if $mathscr{X}$ is a shellable, strongly regular CW sphere or\u0000CW ball of dimension $d$, then for all $lfloor frac{d-1}{2} rfloor leq k\u0000leq d$, [ f_k(mathscr{X}) geq rho(d+1,k)f_d(mathscr{X}) + frac{1}{2}f_k(partial\u0000mathscr{X}), ] with equality precisely when $k=d$ or when $k=d-1$ and\u0000$mathscr{X}$ is simplicial. We further prove that if $mathscr{S}$ is a\u0000strongly regular CW sphere of dimension $d$, and the face poset of\u0000$mathscr{S}$ is both CL-shellable and dual CL-shellable, then\u0000$f_k(mathscr{S}) geq min{f_0(mathscr{S}),f_d(mathscr{S})}$ for all $0\u0000leq k leq d$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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