{"title":"On Diameters of Cayley Graphs over Special Linear Groups","authors":"Eitan Porat","doi":"arxiv-2409.06929","DOIUrl":"https://doi.org/arxiv-2409.06929","url":null,"abstract":"We prove for the matrix group $G=mathrm{SL}_{n}left(mathbb{F}_{p}right)$\u0000that there exist absolute constants $cinleft(0,1right)$ and $C>0$ such that\u0000any symmetric generating set $A$, with $left|Aright|geqleft|Gright|^{1-c}$\u0000has covering number $leq\u0000Cleft(logleft(frac{left|Gright|}{left|Aright|}right)right)^{2}.$ This\u0000result is sharp up to the value of the constant $C>0$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197607","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":"Approximately counting maximal independent set is equivalent to #SAT","authors":"Hao Zhang, Tonghua Su","doi":"arxiv-2409.07035","DOIUrl":"https://doi.org/arxiv-2409.07035","url":null,"abstract":"A maximal independent set is an independent set that is not a subset of any\u0000other independent set. It is also the key problem of mathematics, computer\u0000science, and other fields. A counting problem is a type of computational\u0000problem that associated with the number of solutions. Besides, counting\u0000problems help us better understand several fields such as algorithm analysis,\u0000complexity theory, artificial intelligence, etc. The problem of counting\u0000maximal independent sets is #P-complete. So it is natural to think about\u0000approximate counting for maximal independent sets problem. In this article, we\u0000study the complexity of approximately counting maximal independent sets.\u0000Specifically, we are the first to prove that the #MIS problem is\u0000AP-interreducible with the #SAT of a given general graph.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197614","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":"Nearly optimal coloring of some C4-free graphs","authors":"Ran Chen, Baogang Xu","doi":"arxiv-2409.06944","DOIUrl":"https://doi.org/arxiv-2409.06944","url":null,"abstract":"A class ${cal G}$ of graphs is $chi$-{em polydet} if ${cal G}$ has a\u0000polynomial binding function $f$ and there is a polynomial time algorithm to\u0000determine an $f(omega(G))$-coloring of $Gin {cal G}$. Let $P_t$ and $C_t$\u0000denote a path and a cycle on $t$ vertices, respectively. A {em bull} consists\u0000of a triangle with two disjoint pendant edges, a {em hammer} is obtained by\u0000identifying an end of $P_3$ with a vertex of a triangle, a {em fork$^+$} is\u0000obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a\u0000hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_3, C_4, F)$-free\u0000graphs without clique cutsets and universal cliques, and present a close\u0000relation between $(C_4, F, H)$-free graphs and the Petersen graph. As a\u0000consequence, we show that the classes of $(C_4, F, H)$-free graphs are\u0000$chi$-polydet with nearly optimal linear binding functions.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197605","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":"Problems from BCC30","authors":"Peter J. Cameron","doi":"arxiv-2409.07216","DOIUrl":"https://doi.org/arxiv-2409.07216","url":null,"abstract":"These problems were mostly presented at the problem session at the 30th\u0000British Combinatorial Conference at Queen Mary University of London on 4 July\u00002024. Some were contributed later by conference participants. Thank you to all\u0000the contributors. The problems are given here in alphabetical order of presenter. If no\u0000originator is given, I assume that the presenter is the originator. Please send corrections to me (texttt{pjc20@st-andrews.ac.uk}). Solutions\u0000should be sent to the presenter; I would appreciate a copy too.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197602","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":"Non-empty intersection of longest paths in $P_5$-free and claw-free graphs","authors":"Paloma T. Lima, Amir Nikabadi","doi":"arxiv-2409.07366","DOIUrl":"https://doi.org/arxiv-2409.07366","url":null,"abstract":"A family $mathcal{F}$ of graphs is a textit{Gallai family} if for every\u0000connected graph $Gin mathcal{F}$, all longest paths in $G$ have a common\u0000vertex. While it is not known whether $P_5$-free graphs are a Gallai family,\u0000Long Jr., Milans, and Munaro [The Electronic Journal of Combinatorics, 2023]\u0000showed that this is emph{not} the case for the class of claw-free graphs. We\u0000give a complete characterization of the graphs $H$ of size at most five for\u0000which $(text{claw}, H)$-free graphs form a Gallai family. We also show that\u0000$(P_5, H)$-free graphs form a Gallai family if $H$ is a triangle, a paw, or a\u0000diamond. Both of our results are constructive.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197603","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":"Boosting uniformity in quasirandom groups: fast and simple","authors":"Harm Derksen, Chin Ho Lee, Emanuele Viola","doi":"arxiv-2409.06932","DOIUrl":"https://doi.org/arxiv-2409.06932","url":null,"abstract":"We study the communication complexity of multiplying $ktimes t$ elements\u0000from the group $H=text{SL}(2,q)$ in the number-on-forehead model with $k$\u0000parties. We prove a lower bound of $(tlog H)/c^{k}$. This is an exponential\u0000improvement over previous work, and matches the state-of-the-art in the area. Relatedly, we show that the convolution of $k^{c}$ independent copies of a\u00003-uniform distribution over $H^{m}$ is close to a $k$-uniform distribution.\u0000This is again an exponential improvement over previous work which needed\u0000$c^{k}$ copies. The proofs are remarkably simple; the results extend to other\u0000quasirandom groups. We also show that for any group $H$, any distribution over $H^{m}$ whose\u0000weight-$k$ Fourier coefficients are small is close to a $k$-uniform\u0000distribution. This generalizes previous work in the abelian setting, and the\u0000proof is simpler.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"79 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197615","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}
Imre Bárány, Travis Dillon, Dömötör Pálvölgyi, Dániel Varga
{"title":"Piercing intersecting convex sets","authors":"Imre Bárány, Travis Dillon, Dömötör Pálvölgyi, Dániel Varga","doi":"arxiv-2409.06472","DOIUrl":"https://doi.org/arxiv-2409.06472","url":null,"abstract":"Assume two finite families $mathcal A$ and $mathcal B$ of convex sets in\u0000$mathbb{R}^3$ have the property that $Acap Bne emptyset$ for every $A in\u0000mathcal A$ and $Bin mathcal B$. Is there a constant $gamma >0$ (independent\u0000of $mathcal A$ and $mathcal B$) such that there is a line intersecting\u0000$gamma|mathcal A|$ sets in $mathcal A$ or $gamma|mathcal B|$ sets in\u0000$mathcal B$? This is an intriguing Helly-type question from a paper by\u0000Mart'{i}nez, Roldan and Rubin. We confirm this in the special case when all\u0000sets in $mathcal A$ lie in parallel planes and all sets in $mathcal B$ lie in\u0000parallel planes; in fact, all sets from one of the two families has a line\u0000transversal.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197630","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":"Gallai's Path Decomposition of Levi Graph","authors":"Akankshya Sahu","doi":"arxiv-2409.06298","DOIUrl":"https://doi.org/arxiv-2409.06298","url":null,"abstract":"Gallai's path decomposition conjecture states that for a connected graph $G$\u0000on $n$ vertices, there exist a path decomposition of size $lceil frac{n}{2}\u0000rceil$. Levi graph of order one, denoted by $L_{1}(m,k)$ is a bipartite graph\u0000having vertex partition $(A,B)$, where $A$ is the collection of all $k-1$\u0000subsets of $[m]$ and $B$ is the collection of all $k$ subsets of $[m]$. In this\u0000graph a $k-1$ set is adjacent to a $k$ set if it is properly contained inside\u0000the $k$ set. Path number of a graph $G$ is the minimum size of its path\u0000decomposition. Hence, we can rewrite the Gallai conjecture as the path number\u0000of a connected graph is at most $lceil frac{n}{2} rceil$. In this work we\u0000prove conjecture on $L_{1}(m,k)$ for all $m ge 2 $, $2 le k le m$. Moreover\u0000determines the path number of $L_{1}(m,2)$ for all $m$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225156","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 oriented diameter of graphs with given minimum degree","authors":"Garner Cochran, Zhiyu Wang","doi":"arxiv-2409.06587","DOIUrl":"https://doi.org/arxiv-2409.06587","url":null,"abstract":"ErdH{o}s, Pach, Pollack, and Tuza [J. Combin. Theory Ser. B, 47(1) (1989),\u000073--79] proved that the diameter of a connected $n$-vertex graph with minimum\u0000degree $delta$ is at most $frac{3n}{delta+1}+O(1)$. The oriented diameter of\u0000an undirected graph $G$, denoted by $overrightarrow{diam}(G)$, is the minimum\u0000diameter of a strongly connected orientation of $G$. Bau and Dankelmann\u0000[European J. Combin., 49 (2015), 126--133] showed that for every bridgeless\u0000$n$-vertex graph $G$ with minimum degree $delta$, $overrightarrow{diam}(G)\u0000leq frac{11n}{delta+1}+9$. They also showed an infinite family of graphs\u0000with oriented diameter at least $frac{3n}{delta+1} + O(1)$ and posed the\u0000problem of determining the smallest possible value $c$ for which\u0000$overrightarrow{diam}(G) leq c cdotfrac{3n}{delta+1}+O(1)$ holds. In this\u0000paper, we show that the smallest value $c$ such that the upper bound above\u0000holds for all $deltageq 2$ is $1$, which is best possible.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142197631","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 Generalized Pfaffians","authors":"Jacques Distler, Nathan Donagi, Ron Donagi","doi":"arxiv-2409.06871","DOIUrl":"https://doi.org/arxiv-2409.06871","url":null,"abstract":"The determinant of an anti-symmetric matrix $g$ is the square of its\u0000Pfaffian, which like the determinant is a polynomial in the entries of $g$.\u0000Studies of certain super conformal field theories (of class S) suggested a\u0000conjectural generalization of this, predicting that each of a series of other\u0000polynomials in the entries of $g$ also admit polynomial square roots. Among\u0000other consequences, this conjecture led to a characterization of the local\u0000Hitchin image for type D. Several important special cases had been established\u0000previously. In this paper we prove the conjecture in full.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225148","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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