{"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":"201 1","pages":""},"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
{"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":"194 1","pages":""},"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}
{"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":"6 1","pages":""},"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}
{"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":"7 1","pages":""},"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}
{"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":"50 1","pages":""},"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}
{"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":"69 1","pages":""},"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}
{"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":"33 1","pages":""},"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
{"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":"98 1","pages":""},"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}
{"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":"16 1","pages":""},"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}
{"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":"31 1","pages":""},"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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