{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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}
{"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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