In this note, we establish Andr'{a}sfai--ErdH{o}s--S'{o}s-type stability theorems for two generalized Tur'{a}n problems involving odd cycles, both of which are extensions of the ErdH{o}s Pentagon Problem. Our results strengthen previous results by Lidick'{y}--Murphy~cite{LM21} and Beke--Janzer~cite{BJ24}, while also simplifying parts of their proofs.
{"title":"Generalized Andrásfai--Erdős--Sós theorems for odd cycles","authors":"Zian Chen, Jianfeng Hou, Caiyun Hu, Xizhi Liu","doi":"arxiv-2409.11950","DOIUrl":"https://doi.org/arxiv-2409.11950","url":null,"abstract":"In this note, we establish Andr'{a}sfai--ErdH{o}s--S'{o}s-type stability\u0000theorems for two generalized Tur'{a}n problems involving odd cycles, both of\u0000which are extensions of the ErdH{o}s Pentagon Problem. Our results strengthen\u0000previous results by Lidick'{y}--Murphy~cite{LM21} and\u0000Beke--Janzer~cite{BJ24}, while also simplifying parts of their proofs.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255036","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 $m$ be a positive integer. A group $G$ is said to be an $m$-BCI-group if $G$ has the $k$-BCI property for all positive integers $k$ at most $m$. Let $G$ be a generalized quaternion group of order $4n$ with $ngeq 2$. It is shown that $G$ is a 3-BCI-group if and only if $G$ is a $2$-BCI-group if and only if $n=2$ or $n$ is odd.
{"title":"Isomorphisms of bi-Cayley graphs on generalized quaternion groups","authors":"Jin-Hua Xie","doi":"arxiv-2409.11918","DOIUrl":"https://doi.org/arxiv-2409.11918","url":null,"abstract":"Let $m$ be a positive integer. A group $G$ is said to be an $m$-BCI-group if\u0000$G$ has the $k$-BCI property for all positive integers $k$ at most $m$. Let $G$\u0000be a generalized quaternion group of order $4n$ with $ngeq 2$. It is shown\u0000that $G$ is a 3-BCI-group if and only if $G$ is a $2$-BCI-group if and only if\u0000$n=2$ or $n$ is odd.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255039","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}
In 2014, Vesti proposed the problem of determining the repetition threshold for infinite rich words, i.e., for infinite words in which all factors of length $n$ contain $n$ distinct nonempty palindromic factors. In 2020, Currie, Mol, and Rampersad proved a conjecture of Baranwal and Shallit that the repetition threshold for binary rich words is $2 + sqrt{2}/2$. In this paper, we prove a structure theorem for $16/7$-power-free ternary rich words. Using the structure theorem, we deduce that the repetition threshold for ternary rich words is $1 + 1/(3 - mu) approx 2.25876324$, where $mu$ is the unique real root of the polynomial $x^3 - 2x^2 - 1$.
{"title":"The repetition threshold for ternary rich words","authors":"James D. Currie, Lucas Mol, Jarkko Peltomäki","doi":"arxiv-2409.12068","DOIUrl":"https://doi.org/arxiv-2409.12068","url":null,"abstract":"In 2014, Vesti proposed the problem of determining the repetition threshold\u0000for infinite rich words, i.e., for infinite words in which all factors of\u0000length $n$ contain $n$ distinct nonempty palindromic factors. In 2020, Currie,\u0000Mol, and Rampersad proved a conjecture of Baranwal and Shallit that the\u0000repetition threshold for binary rich words is $2 + sqrt{2}/2$. In this paper,\u0000we prove a structure theorem for $16/7$-power-free ternary rich words. Using\u0000the structure theorem, we deduce that the repetition threshold for ternary rich\u0000words is $1 + 1/(3 - mu) approx 2.25876324$, where $mu$ is the unique real\u0000root of the polynomial $x^3 - 2x^2 - 1$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255037","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 say a directed graph $G$ on $n$ vertices is irredundant if the removal of any edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such that there exists a directed path from $u$ to $v$. We determine the maximum possible number of edges such a graph can have, for every $n in mathbb{N}$. We also characterize the cases of equality. This resolves, in a strong form, a question of Crane and Russell.
{"title":"A note on connectivity in directed graphs","authors":"Stelios Stylianou","doi":"arxiv-2409.12137","DOIUrl":"https://doi.org/arxiv-2409.12137","url":null,"abstract":"We say a directed graph $G$ on $n$ vertices is irredundant if the removal of\u0000any edge reduces the number of ordered pairs of distinct vertices $(u,v)$ such\u0000that there exists a directed path from $u$ to $v$. We determine the maximum\u0000possible number of edges such a graph can have, for every $n in mathbb{N}$.\u0000We also characterize the cases of equality. This resolves, in a strong form, a\u0000question of Crane and Russell.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255034","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}
While every plane triangulation is colourable with three or four colours, Heawood showed that a plane triangulation is 3-colourable if and only if every vertex has even degree. In $d geq 3$ dimensions, however, every $k geq d+1$ may occur as the chromatic number of some triangulation of ${mathbb S}^d$. As a first step, Joswig structurally characterised which triangulations of ${mathbb S}^d$ have a $(d+1)$-colourable 1-skeleton. In the 20 years since Joswig's result, no characterisations have been found for any $k>d+1$. In this paper, we structurally characterise which triangulations of ${mathbb S}^d$ have a $(d+2)$-colourable 1-skeleton: they are precisely the triangulations that have a subdivision such that for every $(d-2)$-cell, the number of incident $(d-1)$-cells is divisible by three.
{"title":"Colouring the 1-skeleton of $d$-dimensional triangulations","authors":"Tim Planken","doi":"arxiv-2409.11762","DOIUrl":"https://doi.org/arxiv-2409.11762","url":null,"abstract":"While every plane triangulation is colourable with three or four colours,\u0000Heawood showed that a plane triangulation is 3-colourable if and only if every\u0000vertex has even degree. In $d geq 3$ dimensions, however, every $k geq d+1$\u0000may occur as the chromatic number of some triangulation of ${mathbb S}^d$. As\u0000a first step, Joswig structurally characterised which triangulations of\u0000${mathbb S}^d$ have a $(d+1)$-colourable 1-skeleton. In the 20 years since\u0000Joswig's result, no characterisations have been found for any $k>d+1$. In this paper, we structurally characterise which triangulations of ${mathbb\u0000S}^d$ have a $(d+2)$-colourable 1-skeleton: they are precisely the\u0000triangulations that have a subdivision such that for every $(d-2)$-cell, the\u0000number of incident $(d-1)$-cells is divisible by three.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268996","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 graph polytopes arising from the vertex weighted graph, which was first introduced and studied by B'ona, Ju, and Yoshida. A conjecture states that for a simple connected graph, the polynomial in the numerator of the Ehrhart series is palindromic. We confirm the conjecture. Furthermore, we introduce the hypergraph polytope. We prove that the simple connected unimodular hypergraph polytopes are integer polytopes. We also prove the polynomial in the numerator of the Ehrhart series of simple connected uniform hypergraph polytopes is palindromic.
顶点加权图是由 B'ona, Ju 和 Yoshida 首次提出和研究的。有一个猜想指出，对于简单相连的图，埃尔哈特数列分子中的多项式是回折的。我们证实了这一猜想。此外，我们还引入了超图多面体。我们证明了简单连接的单模态超图多面体是整数多面体。我们还证明了简单连通均匀超图多面体的埃尔哈特数列分子中的多项式是回折的。
{"title":"Proof of a conjecture on graph polytope","authors":"Feihu Liu","doi":"arxiv-2409.11970","DOIUrl":"https://doi.org/arxiv-2409.11970","url":null,"abstract":"The graph polytopes arising from the vertex weighted graph, which was first\u0000introduced and studied by B'ona, Ju, and Yoshida. A conjecture states that for\u0000a simple connected graph, the polynomial in the numerator of the Ehrhart series\u0000is palindromic. We confirm the conjecture. Furthermore, we introduce the\u0000hypergraph polytope. We prove that the simple connected unimodular hypergraph\u0000polytopes are integer polytopes. We also prove the polynomial in the numerator\u0000of the Ehrhart series of simple connected uniform hypergraph polytopes is\u0000palindromic.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255035","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 Furstenberg set problem, and exceptional set estimate for Marstrand's orthogonal projection in prime fields for all dimensions. We define the Furstenberg index $mathbf{F}(s,t;n,k)$ and the Marstrand index $mathbf{M}(a,s;n,k)$. It is shown that the two-dimensional result for Furstenberg set problem implies all higher dimensional results.
{"title":"Furstenberg set problem and exceptional set estimate in prime fields: dimension two implies higher dimensions","authors":"Shengwen Gan","doi":"arxiv-2409.11637","DOIUrl":"https://doi.org/arxiv-2409.11637","url":null,"abstract":"We study Furstenberg set problem, and exceptional set estimate for\u0000Marstrand's orthogonal projection in prime fields for all dimensions. We define\u0000the Furstenberg index $mathbf{F}(s,t;n,k)$ and the Marstrand index\u0000$mathbf{M}(a,s;n,k)$. It is shown that the two-dimensional result for Furstenberg set problem\u0000implies all higher dimensional results.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255058","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}
A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t.$ We prove that every $2$-connected $[4,2]$-graph of order at least $7$ is pancyclic. This strengthens existing results. There are $2$-connected $[4,2]$-graphs which do not satisfy the Chv'{a}tal-ErdH{o}s condition. We also determine the triangle-free graphs among $[p+2,p]$-graphs for a general $p.$
{"title":"A sufficient condition for pancyclic graphs","authors":"Xingzhi Zhan","doi":"arxiv-2409.11716","DOIUrl":"https://doi.org/arxiv-2409.11716","url":null,"abstract":"A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of\u0000order $s$ has size at least $t.$ We prove that every $2$-connected\u0000$[4,2]$-graph of order at least $7$ is pancyclic. This strengthens existing\u0000results. There are $2$-connected $[4,2]$-graphs which do not satisfy the\u0000Chv'{a}tal-ErdH{o}s condition. We also determine the triangle-free graphs\u0000among $[p+2,p]$-graphs for a general $p.$","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255041","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}
This paper investigates five kinds of systems of $d$-partitions of $[n]$, including symmetric Bollob'{a}s systems, strong Bollob'{a}s systems, Bollob'{a}s systems, skew Bollob'{a}s systems, and weak Bollob'{a}s systems. Many known results on variations of Bollob'{a}s systems are unified. Especially we give a negative answer to a conjecture on Bollob'{a}s systems of $d$-partitions of $[n]$ that was presented by Heged"{u}s and Frankl [European J. Comb., 120 (2024), 103983]. Even though this conjecture does not hold for general Bollob'{a}s systems, we show that it holds for strong Bollob'{a}s systems of $d$-partitions of $[n]$.
{"title":"Variations on Bollobás systems of $d$-partitions","authors":"Yu Fang, Xiaomiao Wang, Tao Feng","doi":"arxiv-2409.11907","DOIUrl":"https://doi.org/arxiv-2409.11907","url":null,"abstract":"This paper investigates five kinds of systems of $d$-partitions of $[n]$,\u0000including symmetric Bollob'{a}s systems, strong Bollob'{a}s systems,\u0000Bollob'{a}s systems, skew Bollob'{a}s systems, and weak Bollob'{a}s systems.\u0000Many known results on variations of Bollob'{a}s systems are unified.\u0000Especially we give a negative answer to a conjecture on Bollob'{a}s systems of\u0000$d$-partitions of $[n]$ that was presented by Heged\"{u}s and Frankl [European\u0000J. Comb., 120 (2024), 103983]. Even though this conjecture does not hold for\u0000general Bollob'{a}s systems, we show that it holds for strong Bollob'{a}s\u0000systems of $d$-partitions of $[n]$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255040","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}
ErdH{o}s and Rado [P. ErdH{o}s, R. Rado, A combinatorial theorem, Journal of the London Mathematical Society 25 (4) (1950) 249-255] introduced the Canonical Ramsey numbers $text{er}(t)$ as the minimum number $n$ such that every edge-coloring of the ordered complete graph $K_n$ contains either a monochromatic, rainbow, upper lexical, or lower lexical clique of order $t$. Richer [D. Richer, Unordered canonical Ramsey numbers, Journal of Combinatorial Theory Series B 80 (2000) 172-177] introduced the unordered asymmetric version of the Canonical Ramsey numbers $text{CR}(s,r)$ as the minimum $n$ such that every edge-coloring of the (unorderd) complete graph $K_n$ contains either a rainbow clique of order $r$, or an orderable clique of order $s$. We show that $text{CR}(s,r) = O(r^3/log r)^{s-2}$, which, up to the multiplicative constant, matches the known lower bound and improves the previously best known bound $text{CR}(s,r) = O(r^3/log r)^{s-1}$ by Jiang [T. Jiang, Canonical Ramsey numbers and proporly colored cycles, Discrete Mathematics 309 (2009) 4247-4252]. We also obtain bounds on the further variant $text{ER}(m,ell,r)$, defined as the minimum $n$ such that every edge-coloring of the (unorderd) complete graph $K_n$ contains either a monochromatic $K_m$, lexical $K_ell$, or rainbow $K_r$.
ErdH{o}s 和 Rado [P.ErdH{o}s, R. Rado, A combinatorial theorem, Journalof the London Mathematical Society 25 (4) (1950) 249-255] 提出了典型拉姆齐数 $/text{er}(t)$，即有序完整图 $K_n$ 的每个边着色都包含阶数为 $t$ 的单色、彩虹、上词性或下词性簇的最小数 $n$ 。Richer [D. Richer, Unordered canonical Ramsey numbers, Journal of CombinatorialTheory Series B 80 (2000) 172-177] 引入了无序非对称版本的 Canonical Ramsey 数 $text{CR}(s,r)$，即使得（无序的）完整图 $K_n$ 的每个边着色包含阶 $r$ 的彩虹簇或阶 $s$ 的可排序簇的最小值 $n$。我们证明，$text{CR}(s,r) = O(r^3/log r)^{s-2}$ 这个值（不含乘法常数）与已知的下界相匹配，并且改进了 Jiang [T.Jiang, Canonical Ramsey numbers and proporly colored cycles, DiscreteMathematics 309 (2009) 4247-4252] 所给出的已知最佳边界 $text{CR}(s,r) = O(r^3//log r)^{s-1}$。我们还得到了进一步变体$text{ER}(m,ell,r)$ 的边界，其定义为：使（无序的）完整图 $K_n$ 的每个边着色都包含单色 $K_m$、词性 $K_ell$ 或彩虹 $K_r$ 的最小值 $n$。
{"title":"On the off-diagonal unordered Erdős-Rado numbers","authors":"Igor Araujo, Dadong Peng","doi":"arxiv-2409.11574","DOIUrl":"https://doi.org/arxiv-2409.11574","url":null,"abstract":"ErdH{o}s and Rado [P. ErdH{o}s, R. Rado, A combinatorial theorem, Journal\u0000of the London Mathematical Society 25 (4) (1950) 249-255] introduced the\u0000Canonical Ramsey numbers $text{er}(t)$ as the minimum number $n$ such that\u0000every edge-coloring of the ordered complete graph $K_n$ contains either a\u0000monochromatic, rainbow, upper lexical, or lower lexical clique of order $t$.\u0000Richer [D. Richer, Unordered canonical Ramsey numbers, Journal of Combinatorial\u0000Theory Series B 80 (2000) 172-177] introduced the unordered asymmetric version\u0000of the Canonical Ramsey numbers $text{CR}(s,r)$ as the minimum $n$ such that\u0000every edge-coloring of the (unorderd) complete graph $K_n$ contains either a\u0000rainbow clique of order $r$, or an orderable clique of order $s$. We show that $text{CR}(s,r) = O(r^3/log r)^{s-2}$, which, up to the\u0000multiplicative constant, matches the known lower bound and improves the\u0000previously best known bound $text{CR}(s,r) = O(r^3/log r)^{s-1}$ by Jiang [T.\u0000Jiang, Canonical Ramsey numbers and proporly colored cycles, Discrete\u0000Mathematics 309 (2009) 4247-4252]. We also obtain bounds on the further variant\u0000$text{ER}(m,ell,r)$, defined as the minimum $n$ such that every edge-coloring\u0000of the (unorderd) complete graph $K_n$ contains either a monochromatic $K_m$,\u0000lexical $K_ell$, or rainbow $K_r$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255056","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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