Pub Date : 2025-08-21DOI: 10.1016/j.aam.2025.102958
Hyunsoo Cho , Byungchan Kim , Eunmi Kim , Ae Ja Yee
Recently, Griffin, Ono, and Tsai examined the distribution of the number of t-hooks in partitions of n, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of t-hooks in self-conjugate partitions of n. Motivated by these studies, in this paper, we further investigate the number of t-hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of t-hooks in doubled distinct partitions and the number of t-shifted hooks in strict partitions. Based on these generating functions, we prove that the number of t-hooks in doubled distinct partitions and the number of t-shifted hooks in strict partitions are both asymptotically normally distributed.
{"title":"On the distribution of t-hooks of doubled distinct partitions","authors":"Hyunsoo Cho , Byungchan Kim , Eunmi Kim , Ae Ja Yee","doi":"10.1016/j.aam.2025.102958","DOIUrl":"10.1016/j.aam.2025.102958","url":null,"abstract":"<div><div>Recently, Griffin, Ono, and Tsai examined the distribution of the number of <em>t</em>-hooks in partitions of <em>n</em>, which was later followed by the work of Craig, Ono, and Singh on the distribution of the number of <em>t</em>-hooks in self-conjugate partitions of <em>n</em>. Motivated by these studies, in this paper, we further investigate the number of <em>t</em>-hooks in some subsets of partitions. More specifically, we obtain the generating functions for the number of <em>t</em>-hooks in doubled distinct partitions and the number of <em>t</em>-shifted hooks in strict partitions. Based on these generating functions, we prove that the number of <em>t</em>-hooks in doubled distinct partitions and the number of <em>t</em>-shifted hooks in strict partitions are both asymptotically normally distributed.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102958"},"PeriodicalIF":1.3,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144879033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-21DOI: 10.1016/j.aam.2025.102953
Christopher Reichling, Leo van Iersel, Yukihiro Murakami
Phylogenetic networks are useful in representing the evolutionary history of taxa. In certain scenarios, one requires a way to compare different networks. In practice, this can be rather difficult, except within specific classes of networks. In this paper, we derive metrics for the class of orchard networks and the class of strongly reticulation-visible networks, from variants of so-called μ-representations, which are vector representations of networks. For both network classes, we impose degree constraints on the vertices, by considering semi-binary networks.
{"title":"Metrics for classes of semi-binary phylogenetic networks using μ-representations","authors":"Christopher Reichling, Leo van Iersel, Yukihiro Murakami","doi":"10.1016/j.aam.2025.102953","DOIUrl":"10.1016/j.aam.2025.102953","url":null,"abstract":"<div><div>Phylogenetic networks are useful in representing the evolutionary history of taxa. In certain scenarios, one requires a way to compare different networks. In practice, this can be rather difficult, except within specific classes of networks. In this paper, we derive metrics for the class of <em>orchard networks</em> and the class of <em>strongly reticulation-visible</em> networks, from variants of so-called <em>μ-representations</em>, which are vector representations of networks. For both network classes, we impose degree constraints on the vertices, by considering <em>semi-binary</em> networks.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102953"},"PeriodicalIF":1.3,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144878156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-19DOI: 10.1016/j.aam.2025.102952
Eva Czabarka , Steven Kelk , Vincent Moulton , László A. Székely
In phylogenetics, a key problem is to construct evolutionary trees from collections of characters where, for a set X of species, a character is simply a function from X onto a set of states. In this context, a key concept is convexity, where a character is convex on a tree with leaf set X if the collection of subtrees spanned by the leaves of the tree that have the same state are pairwise disjoint. Although collections of convex characters on a single tree have been extensively studied over the past few decades, very little is known about coconvex characters, that is, characters that are simultaneously convex on a collection of trees. As a starting point to better understand coconvexity, in this paper we prove a number of extremal results for the following question: What is the minimal number of coconvex characters on a collection of n-leaved trees taken over all collections of size, also if we restrict to coconvex characters which map to k states? As an application of coconvexity, we introduce a new one-parameter family of tree metrics, which range between the coarse Robinson-Foulds distance and the much finer quartet distance. We show that bounds on the quantities in the above question translate into bounds for the diameter of the tree space for the new distances. Our results open up several new interesting directions and questions which have potential applications to, for example, tree spaces and phylogenomics.
{"title":"Coconvex characters on collections of phylogenetic trees","authors":"Eva Czabarka , Steven Kelk , Vincent Moulton , László A. Székely","doi":"10.1016/j.aam.2025.102952","DOIUrl":"10.1016/j.aam.2025.102952","url":null,"abstract":"<div><div>In phylogenetics, a key problem is to construct evolutionary trees from collections of characters where, for a set <em>X</em> of species, a character is simply a function from <em>X</em> onto a set of states. In this context, a key concept is convexity, where a character is convex on a tree with leaf set <em>X</em> if the collection of subtrees spanned by the leaves of the tree that have the same state are pairwise disjoint. Although collections of convex characters on a single tree have been extensively studied over the past few decades, very little is known about <em>coconvex characters</em>, that is, characters that are simultaneously convex on a collection of trees. As a starting point to better understand coconvexity, in this paper we prove a number of extremal results for the following question: <em>What is the minimal number of coconvex characters on a collection of n-leaved trees taken over all collections of size</em> <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span><em>, also if we restrict to coconvex characters which map to k states?</em> As an application of coconvexity, we introduce a new one-parameter family of tree metrics, which range between the coarse Robinson-Foulds distance and the much finer quartet distance. We show that bounds on the quantities in the above question translate into bounds for the diameter of the tree space for the new distances. Our results open up several new interesting directions and questions which have potential applications to, for example, tree spaces and phylogenomics.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"172 ","pages":"Article 102952"},"PeriodicalIF":1.3,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144865208","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-19DOI: 10.1016/j.aam.2025.102959
Laura Pierson
As part of a program to develop K-theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas and , where each of , , and is a family of polynomials that forms a basis for indexed by weak compositions a, and and are monomials in β for each pair of weak compositions. The polynomials are the Lascoux atoms, are the kaons, are the quasiLascoux polynomials, and are the glide polynomials
{"title":"Proof of a K-theoretic polynomial conjecture of Monical, Pechenik, and Searles","authors":"Laura Pierson","doi":"10.1016/j.aam.2025.102959","DOIUrl":"10.1016/j.aam.2025.102959","url":null,"abstract":"<div><div>As part of a program to develop <em>K</em>-theoretic analogues of combinatorially important polynomials, Monical, Pechenik, and Searles (2021) proved two expansion formulas <span><math><msub><mrow><mover><mrow><mi>A</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>b</mi></mrow></msub><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>(</mo><mi>β</mi><mo>)</mo><msub><mrow><mover><mrow><mi>P</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>b</mi></mrow></msub></math></span> and <span><math><msub><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>b</mi></mrow></msub><msubsup><mrow><mi>M</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>(</mo><mi>β</mi><mo>)</mo><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>b</mi></mrow></msub></math></span>, where each of <span><math><msub><mrow><mover><mrow><mi>A</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span>, <span><math><msub><mrow><mover><mrow><mi>P</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span>, <span><math><msub><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span> and <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span> is a family of polynomials that forms a basis for <span><math><mi>Z</mi><mo>[</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo><mo>[</mo><mi>β</mi><mo>]</mo></math></span> indexed by weak compositions <em>a</em>, and <span><math><msubsup><mrow><mi>Q</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>(</mo><mi>β</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>M</mi></mrow><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></msubsup><mo>(</mo><mi>β</mi><mo>)</mo></math></span> are monomials in <em>β</em> for each pair <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span> of weak compositions. The polynomials <span><math><msub><mrow><mover><mrow><mi>A</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span> are the <em><strong>Lascoux atoms</strong></em>, <span><math><msub><mrow><mover><mrow><mi>P</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span> are the <em><strong>kaons</strong></em>, <span><math><msub><mrow><mover><mrow><mi>Q</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span> are the <em><strong>quasiLascoux polynomials</strong></em>, and <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mi>a</mi></mrow></msub></math></span> are the <em><strong>glide polynomials</strong></e","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102959"},"PeriodicalIF":1.3,"publicationDate":"2025-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144864296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-14DOI: 10.1016/j.aam.2025.102957
Patrick Schnider , Pablo Soberón
In a mass partition problem, we are interested in finding equitable partitions of smooth measures in . In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set K. We distinguish the problem when we are allowed to use scaled and translated copies of K and the problem when we are allowed to use scaled isometric copies of K. These problems have only previously been studied if K is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any masses for star-shaped compact sets K with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of K. Additional proofs are included for particular instances of K, such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk–Ulam-type theorems.
{"title":"Cookie cutters: Bisections with fixed shapes","authors":"Patrick Schnider , Pablo Soberón","doi":"10.1016/j.aam.2025.102957","DOIUrl":"10.1016/j.aam.2025.102957","url":null,"abstract":"<div><div>In a mass partition problem, we are interested in finding equitable partitions of smooth measures in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>. In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set <em>K</em>. We distinguish the problem when we are allowed to use scaled and translated copies of <em>K</em> and the problem when we are allowed to use scaled isometric copies of <em>K</em>. These problems have only previously been studied if <em>K</em> is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any <span><math><mi>d</mi><mo>+</mo><mn>1</mn></math></span> masses for star-shaped compact sets <em>K</em> with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of <em>K</em>. Additional proofs are included for particular instances of <em>K</em>, such as hypercubes and cylinders, answering positively a conjecture of Soberón and Takahashi. The proof methods are topological and involve new Borsuk–Ulam-type theorems.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102957"},"PeriodicalIF":1.3,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144830959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-06DOI: 10.1016/j.aam.2025.102951
Robert Cori , Gábor Hetyei
We introduce a Whitney polynomial for hypermaps. For maps, our definition depends only on the underlying graph and coincides with the usual definition, but for general hypermaps it depends on the topological structure. Our invariant satisfies a generalized deletion-contraction recurrence and it may be used to generalize the results of Arratia, Bollobás, Ellis-Monaghan, Martin and Sorkin connecting the circuit partition polynomial to the Martin polynomial of a graph. For hypermaps with hyperedges of length at most three our approach also allows generalizing most results connecting the chromatic polynomial and the flow polynomial with the Tutte polynomial of a graph.
{"title":"A Whitney polynomial for hypermaps","authors":"Robert Cori , Gábor Hetyei","doi":"10.1016/j.aam.2025.102951","DOIUrl":"10.1016/j.aam.2025.102951","url":null,"abstract":"<div><div>We introduce a Whitney polynomial for hypermaps. For maps, our definition depends only on the underlying graph and coincides with the usual definition, but for general hypermaps it depends on the topological structure. Our invariant satisfies a generalized deletion-contraction recurrence and it may be used to generalize the results of Arratia, Bollobás, Ellis-Monaghan, Martin and Sorkin connecting the circuit partition polynomial to the Martin polynomial of a graph. For hypermaps with hyperedges of length at most three our approach also allows generalizing most results connecting the chromatic polynomial and the flow polynomial with the Tutte polynomial of a graph.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102951"},"PeriodicalIF":1.3,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-08-06DOI: 10.1016/j.aam.2025.102950
Yifeng Huang , Ruofan Jiang
We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of . We identity both normal forms as instances of Gröbner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of d “spiral shifting operators” on , which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon [26] and Petrogradsky [25] in terms of the hlex normal form.
{"title":"Lattices in Fq[[T]]d and spiral shifting operators","authors":"Yifeng Huang , Ruofan Jiang","doi":"10.1016/j.aam.2025.102950","DOIUrl":"10.1016/j.aam.2025.102950","url":null,"abstract":"<div><div>We investigate the algebra and combinatorics of an analogue of the Hermite normal form that classifies finite-index submodules of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><msup><mrow><mo>[</mo><mo>[</mo><mi>T</mi><mo>]</mo><mo>]</mo></mrow><mrow><mi>d</mi></mrow></msup></math></span>. We identity both normal forms as instances of Gröbner basis theory under different monomial orders, where the Hermite normal form corresponds to the lex order, and the new normal form the hlex order. We note that the hlex normal form recovers the Smith normal form, a feature not enjoyed by the Hermite normal form. We also identify the combinatorial structure underlying the cell decomposition induced by the hlex normal form, which appears to be of independent interest. Notably, the statistics tracking the cell dimensions is compatible, in a certain way, with a collection of <em>d</em> “spiral shifting operators” on <span><math><msup><mrow><mi>N</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, which pairwise commute and collectively act freely and transitively. Using these operators, we give direct proofs of some new combinatorial identities obtained by translating the results of Solomon <span><span>[26]</span></span> and Petrogradsky <span><span>[25]</span></span> in terms of the hlex normal form.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102950"},"PeriodicalIF":1.3,"publicationDate":"2025-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144780617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-24DOI: 10.1016/j.aam.2025.102940
Michael Cuntz , Thorsten Holm , Peter Jørgensen
This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a T-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Çanakçı and Jørgensen.
{"title":"Non-commutative friezes and their determinants, the non-commutative Laurent phenomenon for weak friezes, and frieze gluing","authors":"Michael Cuntz , Thorsten Holm , Peter Jørgensen","doi":"10.1016/j.aam.2025.102940","DOIUrl":"10.1016/j.aam.2025.102940","url":null,"abstract":"<div><div>This paper studies a non-commutative generalisation of Coxeter friezes due to Berenstein and Retakh. It generalises several earlier results to this situation: A formula for frieze determinants, a <em>T</em>-path formula expressing the Laurent phenomenon, and results on gluing friezes together. One of our tools is a non-commutative version of the weak friezes introduced by Çanakçı and Jørgensen.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102940"},"PeriodicalIF":1.0,"publicationDate":"2025-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144696533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-23DOI: 10.1016/j.aam.2025.102938
Giuseppe Cotardo , Alberto Ravagnani , Ferdinando Zullo
We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.
{"title":"Whitney numbers of rank-metric lattices and code enumeration","authors":"Giuseppe Cotardo , Alberto Ravagnani , Ferdinando Zullo","doi":"10.1016/j.aam.2025.102938","DOIUrl":"10.1016/j.aam.2025.102938","url":null,"abstract":"<div><div>We investigate the Whitney numbers of the first kind of rank-metric lattices, which are closely linked to the open problem of enumerating rank-metric codes having prescribed parameters. We apply methods from the theory of hyperovals and linear sets to compute these Whitney numbers for infinite families of rank-metric lattices. As an application of our results, we prove asymptotic estimates on the density function of certain rank-metric codes that have been conjectured in previous work.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102938"},"PeriodicalIF":1.0,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-07-18DOI: 10.1016/j.aam.2025.102939
Fan Chung , Qizhong Lin
<div><div>For graphs <em>G</em> and <em>H</em>, we consider Ramsey numbers <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> with tight lower bounds, namely, <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the chromatic number of <em>G</em> and <span><math><mo>|</mo><mi>H</mi><mo>|</mo></math></span> denotes the number of vertices in <em>H</em>. We say <em>H</em> is <em>G</em>-good if the equality holds.</div><div>Let <span><math><mi>G</mi><mo>+</mo><mi>H</mi></math></span> be the join graph obtained from graphs <em>G</em> and <em>H</em> by adding all edges between the disjoint vertex sets of <em>G</em> and <em>H</em>. Let <em>nH</em> denote the union graph of <em>n</em> disjoint copies of <em>H</em>. We show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><mi>H</mi></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <em>n</em> is sufficiently large. In particular, the fan-graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <span><math><mi>n</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, improving previous tower-type lower bounds for <em>n</em> due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>F</mi><mo>)</mo></math></span> for the case of <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>, the complete <em>p</em>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2023), we show that for any fixed graph <em>H</em>,<span><span><span><math><mrow><mi>r</mi><mo>(</mo><mi>G</m
{"title":"Fan-complete Ramsey numbers","authors":"Fan Chung , Qizhong Lin","doi":"10.1016/j.aam.2025.102939","DOIUrl":"10.1016/j.aam.2025.102939","url":null,"abstract":"<div><div>For graphs <em>G</em> and <em>H</em>, we consider Ramsey numbers <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> with tight lower bounds, namely, <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo><mo>≥</mo><mo>(</mo><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mo>|</mo><mi>H</mi><mo>|</mo><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>χ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes the chromatic number of <em>G</em> and <span><math><mo>|</mo><mi>H</mi><mo>|</mo></math></span> denotes the number of vertices in <em>H</em>. We say <em>H</em> is <em>G</em>-good if the equality holds.</div><div>Let <span><math><mi>G</mi><mo>+</mo><mi>H</mi></math></span> be the join graph obtained from graphs <em>G</em> and <em>H</em> by adding all edges between the disjoint vertex sets of <em>G</em> and <em>H</em>. Let <em>nH</em> denote the union graph of <em>n</em> disjoint copies of <em>H</em>. We show that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><mi>H</mi></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <em>n</em> is sufficiently large. In particular, the fan-graph <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>n</mi><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-good if <span><math><mi>n</mi><mo>=</mo><mi>Ω</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, improving previous tower-type lower bounds for <em>n</em> due to Li and Rousseau (1996). Moreover, we give a stronger lower bound inequality for Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>F</mi><mo>)</mo></math></span> for the case of <span><math><mi>G</mi><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>, the complete <em>p</em>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>. In particular, using a stability-supersaturation lemma by Fox, He and Wigderson (2023), we show that for any fixed graph <em>H</em>,<span><span><span><math><mrow><mi>r</mi><mo>(</mo><mi>G</m","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"171 ","pages":"Article 102939"},"PeriodicalIF":1.0,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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