We prove for the matrix group $G=mathrm{SL}_{n}left(mathbb{F}_{p}right)$ that there exist absolute constants $cinleft(0,1right)$ and $C>0$ such that any symmetric generating set $A$, with $left|Aright|geqleft|Gright|^{1-c}$ has covering number $leq Cleft(logleft(frac{left|Gright|}{left|Aright|}right)right)^{2}.$ This result is sharp up to the value of the constant $C>0$.
{"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}
A maximal independent set is an independent set that is not a subset of any other independent set. It is also the key problem of mathematics, computer science, and other fields. A counting problem is a type of computational problem that associated with the number of solutions. Besides, counting problems help us better understand several fields such as algorithm analysis, complexity theory, artificial intelligence, etc. The problem of counting maximal independent sets is #P-complete. So it is natural to think about approximate counting for maximal independent sets problem. In this article, we study the complexity of approximately counting maximal independent sets. Specifically, we are the first to prove that the #MIS problem is AP-interreducible with the #SAT of a given general graph.
{"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}
A class ${cal G}$ of graphs is $chi$-{em polydet} if ${cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(omega(G))$-coloring of $Gin {cal G}$. Let $P_t$ and $C_t$ denote a path and a cycle on $t$ vertices, respectively. A {em bull} consists of a triangle with two disjoint pendant edges, a {em hammer} is obtained by identifying an end of $P_3$ with a vertex of a triangle, a {em fork$^+$} is obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_3, C_4, F)$-free graphs without clique cutsets and universal cliques, and present a close relation between $(C_4, F, H)$-free graphs and the Petersen graph. As a consequence, we show that the classes of $(C_4, F, H)$-free graphs are $chi$-polydet with nearly optimal linear binding functions.
{"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}
These problems were mostly presented at the problem session at the 30th British Combinatorial Conference at Queen Mary University of London on 4 July 2024. Some were contributed later by conference participants. Thank you to all the contributors. The problems are given here in alphabetical order of presenter. If no originator is given, I assume that the presenter is the originator. Please send corrections to me (texttt{pjc20@st-andrews.ac.uk}). Solutions should be sent to the presenter; I would appreciate a copy too.
{"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}
A family $mathcal{F}$ of graphs is a textit{Gallai family} if for every connected graph $Gin mathcal{F}$, all longest paths in $G$ have a common vertex. While it is not known whether $P_5$-free graphs are a Gallai family, Long Jr., Milans, and Munaro [The Electronic Journal of Combinatorics, 2023] showed that this is emph{not} the case for the class of claw-free graphs. We give a complete characterization of the graphs $H$ of size at most five for which $(text{claw}, H)$-free graphs form a Gallai family. We also show that $(P_5, H)$-free graphs form a Gallai family if $H$ is a triangle, a paw, or a diamond. Both of our results are constructive.
{"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}
We study the communication complexity of multiplying $ktimes t$ elements from the group $H=text{SL}(2,q)$ in the number-on-forehead model with $k$ parties. We prove a lower bound of $(tlog H)/c^{k}$. This is an exponential improvement 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 3-uniform distribution over $H^{m}$ is close to a $k$-uniform distribution. This is again an exponential improvement over previous work which needed $c^{k}$ copies. The proofs are remarkably simple; the results extend to other quasirandom groups. We also show that for any group $H$, any distribution over $H^{m}$ whose weight-$k$ Fourier coefficients are small is close to a $k$-uniform distribution. This generalizes previous work in the abelian setting, and the proof is simpler.
{"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
Assume two finite families $mathcal A$ and $mathcal B$ of convex sets in $mathbb{R}^3$ have the property that $Acap Bne emptyset$ for every $A in mathcal A$ and $Bin mathcal B$. Is there a constant $gamma >0$ (independent of $mathcal A$ and $mathcal B$) such that there is a line intersecting $gamma|mathcal A|$ sets in $mathcal A$ or $gamma|mathcal B|$ sets in $mathcal B$? This is an intriguing Helly-type question from a paper by Mart'{i}nez, Roldan and Rubin. We confirm this in the special case when all sets in $mathcal A$ lie in parallel planes and all sets in $mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.
{"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}
Gallai's path decomposition conjecture states that for a connected graph $G$ on $n$ vertices, there exist a path decomposition of size $lceil frac{n}{2} rceil$. Levi graph of order one, denoted by $L_{1}(m,k)$ is a bipartite graph having vertex partition $(A,B)$, where $A$ is the collection of all $k-1$ subsets of $[m]$ and $B$ is the collection of all $k$ subsets of $[m]$. In this graph a $k-1$ set is adjacent to a $k$ set if it is properly contained inside the $k$ set. Path number of a graph $G$ is the minimum size of its path decomposition. Hence, we can rewrite the Gallai conjecture as the path number of a connected graph is at most $lceil frac{n}{2} rceil$. In this work we prove conjecture on $L_{1}(m,k)$ for all $m ge 2 $, $2 le k le m$. Moreover determines the path number of $L_{1}(m,2)$ for all $m$.
{"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}
ErdH{o}s, Pach, Pollack, and Tuza [J. Combin. Theory Ser. B, 47(1) (1989), 73--79] proved that the diameter of a connected $n$-vertex graph with minimum degree $delta$ is at most $frac{3n}{delta+1}+O(1)$. The oriented diameter of an undirected graph $G$, denoted by $overrightarrow{diam}(G)$, is the minimum diameter of a strongly connected orientation of $G$. Bau and Dankelmann [European J. Combin., 49 (2015), 126--133] showed that for every bridgeless $n$-vertex graph $G$ with minimum degree $delta$, $overrightarrow{diam}(G) leq frac{11n}{delta+1}+9$. They also showed an infinite family of graphs with oriented diameter at least $frac{3n}{delta+1} + O(1)$ and posed the problem of determining the smallest possible value $c$ for which $overrightarrow{diam}(G) leq c cdotfrac{3n}{delta+1}+O(1)$ holds. In this paper, we show that the smallest value $c$ such that the upper bound above holds for all $deltageq 2$ is $1$, which is best possible.
{"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}
The determinant of an anti-symmetric matrix $g$ is the square of its Pfaffian, which like the determinant is a polynomial in the entries of $g$. Studies of certain super conformal field theories (of class S) suggested a conjectural generalization of this, predicting that each of a series of other polynomials in the entries of $g$ also admit polynomial square roots. Among other consequences, this conjecture led to a characterization of the local Hitchin image for type D. Several important special cases had been established previously. In this paper we prove the conjecture in full.
对某些超共形场论（S 类）的研究提出了对这一点的猜想性概括，预言在 $g$ 的条目中的一系列其他多项式中，每个多项式都有多项式平方根。这一猜想还带来了其他后果，其中包括对 D 型的局部希钦象的描述。在本文中，我们将全面证明这一猜想。
{"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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