Kochol introduced the assigning polynomial $F(G,alpha;k)$ to count nowhere-zero $(A,b)$-flows of a graph $G$, where $A$ is a finite Abelian group and $alpha$ is a ${0,1}$-assigning from a family $Lambda(G)$ of certain nonempty vertex subsets of $G$ to ${0,1}$. We introduce the concepts of $b$-compatible graph and $b$-compatible broken bond to give an explicit formula for the assigning polynomials and to examine their coefficients. More specifically, for a function $b:V(G)to A$, let $alpha_{G,b}$ be a ${0,1}$-assigning of $G$ such that for each $XinLambda(G)$, $alpha_{G,b}(X)=0$ if and only if $sum_{vin X}b(v)=0$. We show that for any ${0,1}$-assigning $alpha$ of $G$, if there exists a function $b:V(G)to A$ such that $G$ is $b$-compatible and $alpha=alpha_{G,b}$, then the assigning polynomial $F(G,alpha;k)$ has the $b$-compatible spanning subgraph expansion [ F(G,alpha;k)=sum_{substack{Ssubseteq E(G),G-Smbox{ is $b$-compatible}}}(-1)^{|S|}k^{m(G-S)}, ] and is the following form $F(G,alpha;k)=sum_{i=0}^{m(G)}(-1)^ia_i(G,alpha)k^{m(G)-i}$, where each $a_i(G,alpha)$ is the number of subsets $S$ of $E(G)$ having $i$ edges such that $G-S$ is $b$-compatible and $S$ contains no $b$-compatible broken bonds with respect to a total order on $E(G)$. Applying the counting interpretation, we also obtain unified comparison relations for the signless coefficients of assigning polynomials. Namely, for any ${0,1}$-assignings $alpha,alpha'$ of $G$, if there exist functions $b:V(G)to A$ and $b':V(G)to A'$ such that $G$ is both $b$-compatible and $b'$-compatible, $alpha=alpha_{G,b}$, $alpha'=alpha_{G,b'}$ and $alpha(X)lealpha'(X)$ for all $XinLambda(G)$, then [ a_i(G,alpha)le a_i(G,alpha') quad mbox{ for }quad i=0,1,ldots, m(G). ]
Kochol 引入了赋值多项式 $F(G,alpha;k)$来计算图 $G$ 的无处为零的 $(A,b)$流，其中 $A$ 是一个有限阿贝尔群，$alpha$ 是一个从 $G$ 的某些非空顶点子集的族 $Lambda(G)$ 到 ${0,1}$的 ${0,1}$赋值。我们引入了$b$相容图和$b$相容断键的概念，给出了赋值多项式的明确公式，并检验了它们的系数。更具体地说，对于一个函数 $b:V(G)to A$，让 $alpha_{G,b}$ 是 $G$ 的一个 ${0,1}$赋值，使得对于每个 $XinLambda(G)$, $alpha_{G,b}(X)=0$ if and only if $sum_{vin X}b(v)=0$.我们证明，对于 $G$ 的任意${0,1}$赋值 $alpha$，如果存在一个函数 $b：V(G)to A$，使得 $G$ 是 $b$ 兼容的，并且 $alpha=alpha_{G,b}$ ，那么赋值多项式 $F(G,alpha;k)$ 具有 $b$ 兼容的跨子图展开图（[F(G,alpha；k)=sum_{substack{Ssubseteq E(G),G-Smbox{ is$b$-compatible}}(-1)^{|S|}k^{m(G-S)}, ]并且是下面的形式$F(G,alpha；k)=sum_{i=0}^{m(G)}(-1)^ia_i(G,alpha)k^{m(G)-i}$，其中每个$a_i(G,alpha)$是$E(G)$中具有$i$边的子集$S$的个数，使得$G-S$是$b$兼容的，并且相对于$E(G)$上的总阶，$S$不包含任何$b$兼容的断键。应用计数解释，我们还得到了赋值多项式无符号系数的统一比较关系。也就是说，对于 $G$ 的任意 ${0,1}$ 分配 $alpha,alpha'$，如果存在函数 $b:V(G)to A$ 和 $b'：这样，$G$既与$b$兼容又与$b'兼容，$alpha=alpha_{G,b}$,$alpha'=alpha_{G、b'}$ and $alpha(X)lealpha'(X)$ for all $XinLambda(G)$,then [ a_i(G,alpha)le a_i(G,alpha') quad mbox{ for }quad i=0,1,ldots,m(G).
{"title":"Counting Flows of $b$-compatible Graphs","authors":"Houshan Fu, Xiangyu Ren, Suijie Wang","doi":"arxiv-2409.09634","DOIUrl":"https://doi.org/arxiv-2409.09634","url":null,"abstract":"Kochol introduced the assigning polynomial $F(G,alpha;k)$ to count\u0000nowhere-zero $(A,b)$-flows of a graph $G$, where $A$ is a finite Abelian group\u0000and $alpha$ is a ${0,1}$-assigning from a family $Lambda(G)$ of certain\u0000nonempty vertex subsets of $G$ to ${0,1}$. We introduce the concepts of\u0000$b$-compatible graph and $b$-compatible broken bond to give an explicit formula\u0000for the assigning polynomials and to examine their coefficients. More\u0000specifically, for a function $b:V(G)to A$, let $alpha_{G,b}$ be a\u0000${0,1}$-assigning of $G$ such that for each $XinLambda(G)$,\u0000$alpha_{G,b}(X)=0$ if and only if $sum_{vin X}b(v)=0$. We show that for any\u0000${0,1}$-assigning $alpha$ of $G$, if there exists a function $b:V(G)to A$\u0000such that $G$ is $b$-compatible and $alpha=alpha_{G,b}$, then the assigning\u0000polynomial $F(G,alpha;k)$ has the $b$-compatible spanning subgraph expansion\u0000[ F(G,alpha;k)=sum_{substack{Ssubseteq E(G),G-Smbox{ is\u0000$b$-compatible}}}(-1)^{|S|}k^{m(G-S)}, ] and is the following form\u0000$F(G,alpha;k)=sum_{i=0}^{m(G)}(-1)^ia_i(G,alpha)k^{m(G)-i}$, where each\u0000$a_i(G,alpha)$ is the number of subsets $S$ of $E(G)$ having $i$ edges such\u0000that $G-S$ is $b$-compatible and $S$ contains no $b$-compatible broken bonds\u0000with respect to a total order on $E(G)$. Applying the counting interpretation,\u0000we also obtain unified comparison relations for the signless coefficients of\u0000assigning polynomials. Namely, for any ${0,1}$-assignings $alpha,alpha'$ of\u0000$G$, if there exist functions $b:V(G)to A$ and $b':V(G)to A'$ such that $G$\u0000is both $b$-compatible and $b'$-compatible, $alpha=alpha_{G,b}$,\u0000$alpha'=alpha_{G,b'}$ and $alpha(X)lealpha'(X)$ for all $XinLambda(G)$,\u0000then [ a_i(G,alpha)le a_i(G,alpha') quad mbox{ for }quad i=0,1,ldots,\u0000m(G). ]","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255076","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}
Jayakumar C, Sreekumar K. G., Manilal K., Ismail Naci Cangul
Let $mathscr{B}_n = { pm x_1, pm x_2, pm x_3, cdots, pm x_{n-1}, x_n }$ where $n>1$ is fixed, $x_i in mathbb{R}^+$, $i = 1, 2, 3, cdots, n$ and $x_1 < x_2 < x_3 < cdots < x_n$. Let $phi(mathscr{B}_n)$ be the set of all non-empty subsets $S = {u_1, u_2,cdots, u_t}$ of $mathscr{B}_n$ such that $|u_1|<|u_2|
{"title":"Invariants of Bipartite Kneser B type-MakeLowercase{k} graphs","authors":"Jayakumar C, Sreekumar K. G., Manilal K., Ismail Naci Cangul","doi":"arxiv-2409.09317","DOIUrl":"https://doi.org/arxiv-2409.09317","url":null,"abstract":"Let $mathscr{B}_n = { pm x_1, pm x_2, pm x_3, cdots, pm x_{n-1}, x_n\u0000}$ where $n>1$ is fixed, $x_i in mathbb{R}^+$, $i = 1, 2, 3, cdots, n$ and\u0000$x_1 < x_2 < x_3 < cdots < x_n$. Let $phi(mathscr{B}_n)$ be the set of all\u0000non-empty subsets $S = {u_1, u_2,cdots, u_t}$ of $mathscr{B}_n$ such that\u0000$|u_1|<|u_2|<cdots <|u_{t-1}|<u_t $ where $u_tin mathbb{R}^+$. Let\u0000$mathscr{B}_n^+ = { x_1, x_2, x_3, cdots, x_{n-1}, x_n }$. For a fixed $k$,\u0000let $V_1$ be the set of $k$-element subsets of $mathscr{B}_n^+$, $1 leq k\u0000<n$. $V_2= phi(mathscr{B}_n)-V_1$. For any $A in V_2$, let $A^dagger =\u0000{lvert x rvert: x in A}$. Define a bipartite graph with parts $V_1$ and\u0000$V_2$ and having adjacency as $X in V_1$ is adjacent to $Yin V_2$ if and only\u0000if $X subset Y^dagger$ or $Y^dagger subset X$. A graph of this type is\u0000called a bipartite Kneser B type-$k$ graph and denoted by $H_B(n,k)$. In this\u0000paper, we calculated various graph invariants of $H_B(n,k)$.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254774","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 prove that for every complete graph $K_t$, all graphs $G$ with no induced subgraph isomorphic to a subdivision of $K_t$ have a stable subset of size at least $|G|/{rm polylog}|G|$. This is close to best possible, because for $tge 6$, not all such graphs $G$ have a stable set of linear size, even if $G$ is triangle-free.
{"title":"Subdivisions and near-linear stable sets","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"arxiv-2409.09400","DOIUrl":"https://doi.org/arxiv-2409.09400","url":null,"abstract":"We prove that for every complete graph $K_t$, all graphs $G$ with no induced\u0000subgraph isomorphic to a subdivision of $K_t$ have a stable subset of size at\u0000least $|G|/{rm polylog}|G|$. This is close to best possible, because for $tge\u00006$, not all such graphs $G$ have a stable set of linear size, even if $G$ is\u0000triangle-free.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255079","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}
When $H$ is a forest, the Gy'arf'as-Sumner conjecture implies that every graph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique number has a stable set of linear size. We cannot prove that, but we prove that every such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a forest, there need not be such a stable set. Second, we prove that when $H$ is a ``multibroom'', there {em is} a stable set of linear size. As a consequence, we deduce that all multibrooms satisfy a ``fractional colouring'' version of the Gy'arf'as-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.
{"title":"Trees and near-linear stable sets","authors":"Tung Nguyen, Alex Scott, Paul Seymour","doi":"arxiv-2409.09397","DOIUrl":"https://doi.org/arxiv-2409.09397","url":null,"abstract":"When $H$ is a forest, the Gy'arf'as-Sumner conjecture implies that every\u0000graph $G$ with no induced subgraph isomorphic to $H$ and with bounded clique\u0000number has a stable set of linear size. We cannot prove that, but we prove that\u0000every such graph $G$ has a stable set of size $|G|^{1-o(1)}$. If $H$ is not a\u0000forest, there need not be such a stable set. Second, we prove that when $H$ is a ``multibroom'', there {em is} a stable\u0000set of linear size. As a consequence, we deduce that all multibrooms satisfy a\u0000``fractional colouring'' version of the Gy'arf'as-Sumner conjecture. Finally, we discuss extensions of our results to the multicolour setting.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254776","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 a family of Riordan arrays whose square symmetrizations lead to the Robbins numbers as well as numbers associated to the $20$ vertex model. We provide closed-form expressions for the elements of these arrays, and also give a canonical Catalan factorization for them. We describe a related family of Riordan arrays whose symmetrizations also lead to the same integer sequences.
{"title":"A Riordan array family for some integrable lattice models","authors":"Paul Barry","doi":"arxiv-2409.09547","DOIUrl":"https://doi.org/arxiv-2409.09547","url":null,"abstract":"We study a family of Riordan arrays whose square symmetrizations lead to the\u0000Robbins numbers as well as numbers associated to the $20$ vertex model. We\u0000provide closed-form expressions for the elements of these arrays, and also give\u0000a canonical Catalan factorization for them. We describe a related family of\u0000Riordan arrays whose symmetrizations also lead to the same integer sequences.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255077","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 $mathfrak{p} subset V$ be a polytope and $xi in V_{mathbb{C}}^*$. We obtain an expression for $I(mathfrak{p}; alpha) := int_{mathfrak{p}} e^{langle alpha, x rangle} dx$ as a sum of meromorphic functions in $alpha in V^*_{mathbb{C}}$ parametrized by the faces $mathfrak{f}$ of $mathfrak{p}$ on which $langle xi, x rangle$ is constant. Each term only depends on the local geometry of $mathfrak{p}$ near $mathfrak{f}$ (and on $xi$) and is holomorphic at $alpha = xi$. When $langle xi, cdot rangle$ is only constant on the vertices of $mathfrak{p}$ our formula reduces to Brion's formula. Suppose $mathfrak{p}$ is a rational polytope with respect to a lattice $Lambda$. We obtain an expression for $S(mathfrak{p}; alpha) := sum_{lambda in mathfrak{p} cap Lambda} e^{langle alpha, lambda rangle}$ as a sum of meromorphic functions parametrized by the faces $mathfrak{f}$ on which $e^{langle xi, x rangle} = 1$ on a finite index sublattice of $text{lin}(mathfrak{f}) cap Lambda$. Each term only depends on the local geometry of $mathfrak{p}$ near $mathfrak{f}$ (and on $xi$ and $Lambda$) and is holomorphic at $alpha = xi$. When $e^{langle xi, cdot rangle} neq 1$ at any non-zero lattice point on a line through the origin parallel to an edge of $mathfrak{p}$, our formula reduces to Brion's formula, and when $xi = 0$, it reduces to the Ehrhart quasi-polynomial. Our formulas are particularly useful for understanding how $I(mathfrak{p}(h); xi)$ and $S(mathfrak{p}(h); xi)$ vary in a family of polytopes $mathfrak{p}(h)$ with the same normal fan. When considering dilates of a fixed polytope, our formulas may be viewed as polytopal analogues of Laplace's method and the method of stationary phase. Such expressions naturally show up in analysis on symmetric spaces and affine buildings.
{"title":"A degenerate version of Brion's formula","authors":"Carsten Peterson","doi":"arxiv-2409.09544","DOIUrl":"https://doi.org/arxiv-2409.09544","url":null,"abstract":"Let $mathfrak{p} subset V$ be a polytope and $xi in V_{mathbb{C}}^*$. We\u0000obtain an expression for $I(mathfrak{p}; alpha) := int_{mathfrak{p}}\u0000e^{langle alpha, x rangle} dx$ as a sum of meromorphic functions in $alpha\u0000in V^*_{mathbb{C}}$ parametrized by the faces $mathfrak{f}$ of\u0000$mathfrak{p}$ on which $langle xi, x rangle$ is constant. Each term only\u0000depends on the local geometry of $mathfrak{p}$ near $mathfrak{f}$ (and on\u0000$xi$) and is holomorphic at $alpha = xi$. When $langle xi, cdot rangle$\u0000is only constant on the vertices of $mathfrak{p}$ our formula reduces to\u0000Brion's formula. Suppose $mathfrak{p}$ is a rational polytope with respect to a lattice\u0000$Lambda$. We obtain an expression for $S(mathfrak{p}; alpha) :=\u0000sum_{lambda in mathfrak{p} cap Lambda} e^{langle alpha, lambda\u0000rangle}$ as a sum of meromorphic functions parametrized by the faces\u0000$mathfrak{f}$ on which $e^{langle xi, x rangle} = 1$ on a finite index\u0000sublattice of $text{lin}(mathfrak{f}) cap Lambda$. Each term only depends\u0000on the local geometry of $mathfrak{p}$ near $mathfrak{f}$ (and on $xi$ and\u0000$Lambda$) and is holomorphic at $alpha = xi$. When $e^{langle xi, cdot\u0000rangle} neq 1$ at any non-zero lattice point on a line through the origin\u0000parallel to an edge of $mathfrak{p}$, our formula reduces to Brion's formula,\u0000and when $xi = 0$, it reduces to the Ehrhart quasi-polynomial. Our formulas are particularly useful for understanding how\u0000$I(mathfrak{p}(h); xi)$ and $S(mathfrak{p}(h); xi)$ vary in a family of\u0000polytopes $mathfrak{p}(h)$ with the same normal fan. When considering dilates\u0000of a fixed polytope, our formulas may be viewed as polytopal analogues of\u0000Laplace's method and the method of stationary phase. Such expressions naturally\u0000show up in analysis on symmetric spaces and affine buildings.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"70 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255078","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 prove that with probability $1-o(1)$ as $n to infty$, a uniformly random Latin square of order $n$ contains no subsquare of order $4$ or more, resolving a conjecture of McKay and Wanless. We also show that the expected number of subsquares of order 3 is bounded.
{"title":"Subsquares in random Latin squares","authors":"Jack Allsop, Ian M. Wanless","doi":"arxiv-2409.08446","DOIUrl":"https://doi.org/arxiv-2409.08446","url":null,"abstract":"We prove that with probability $1-o(1)$ as $n to infty$, a uniformly random\u0000Latin square of order $n$ contains no subsquare of order $4$ or more, resolving\u0000a conjecture of McKay and Wanless. We also show that the expected number of\u0000subsquares of order 3 is bounded.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254780","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 classical result of Kleitman determines the maximum number $f(n,s)$ of subsets in a family $mathcal{F}subseteq 2^{[n]}$ of sets that do not contain distinct sets $F_1,F_2,dots,F_s$ that are pairwise disjoint in the case $nequiv 0,-1$ (mod $s$). Katona and Nagy determined the maximum size of a family of subsets of an $n$-element set that does not contain $A_1,A_2,dots,A_t,B_1,B_2,dots,B_t$ with $bigcup_{i=1}^t A_i$ and $bigcup_{i=1}^t B_i$ being disjoint. In this paper, we consider the problem of finding the maximum number $vex(n,K_{stimes t})$ in a family $mathcal{F}subseteq 2^{[n]}$ without sets $F^1_1,dots,F^1_t,dots,F^s_1,dots,F^s_t$ such that $G_j=bigcup_{i=1}^tF^j_i$ $j=1,2,dots,s$ are pairwise disjoint. We determine the asymptotics of $2^n-vex(n,K_{stimes t})$ if $nequiv -1$ (mod $s$) for all $t$, and if $nequiv 0$ (mod $s$), $tge 3$ and show that in this latter case the asymptotics of the $t=2$ subcase is different from both the $t=1$ and $tge 3$ subcases.
{"title":"On a generalization of a result of Kleitman","authors":"Ryan R. Martin, Balázs Patkós","doi":"arxiv-2409.08694","DOIUrl":"https://doi.org/arxiv-2409.08694","url":null,"abstract":"A classical result of Kleitman determines the maximum number $f(n,s)$ of\u0000subsets in a family $mathcal{F}subseteq 2^{[n]}$ of sets that do not contain\u0000distinct sets $F_1,F_2,dots,F_s$ that are pairwise disjoint in the case\u0000$nequiv 0,-1$ (mod $s$). Katona and Nagy determined the maximum size of a\u0000family of subsets of an $n$-element set that does not contain\u0000$A_1,A_2,dots,A_t,B_1,B_2,dots,B_t$ with $bigcup_{i=1}^t A_i$ and\u0000$bigcup_{i=1}^t B_i$ being disjoint. In this paper, we consider the problem of\u0000finding the maximum number $vex(n,K_{stimes t})$ in a family\u0000$mathcal{F}subseteq 2^{[n]}$ without sets\u0000$F^1_1,dots,F^1_t,dots,F^s_1,dots,F^s_t$ such that\u0000$G_j=bigcup_{i=1}^tF^j_i$ $j=1,2,dots,s$ are pairwise disjoint. We determine\u0000the asymptotics of $2^n-vex(n,K_{stimes t})$ if $nequiv -1$ (mod $s$) for all\u0000$t$, and if $nequiv 0$ (mod $s$), $tge 3$ and show that in this latter case\u0000the asymptotics of the $t=2$ subcase is different from both the $t=1$ and $tge\u00003$ subcases.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254782","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}
Christopher Sumnicht, Jamison W. Weber, Dhanush R. Giriyan, Arunabha Sen
Significant work has been done on computing the ``average'' optimal solution value for various $mathsf{NP}$-complete problems using the Erd"{o}s-R'{e}nyi model to establish emph{critical thresholds}. Critical thresholds define narrow bounds for the optimal solution of a problem instance such that the probability that the solution value lies outside these bounds vanishes as the instance size approaches infinity. In this paper, we extend the Erd"{o}s-R'{e}nyi model to general hypergraphs on $n$ vertices and $M$ hyperedges. We consider the problem of determining critical thresholds for the largest cardinality matching, and we show that for $M=o(1.155^n)$ the size of the maximum cardinality matching is almost surely 1. On the other hand, if $M=Theta(2^n)$ then the size of the maximum cardinality matching is $Omega(n^{frac12-gamma})$ for an arbitrary $gamma >0$. Lastly, we address the gap where $Omega(1.155^n)=M=o(2^n)$ empirically through computer simulations.
{"title":"Critical Thresholds for Maximum Cardinality Matching on General Hypergraphs","authors":"Christopher Sumnicht, Jamison W. Weber, Dhanush R. Giriyan, Arunabha Sen","doi":"arxiv-2409.09155","DOIUrl":"https://doi.org/arxiv-2409.09155","url":null,"abstract":"Significant work has been done on computing the ``average'' optimal solution\u0000value for various $mathsf{NP}$-complete problems using the Erd\"{o}s-R'{e}nyi\u0000model to establish emph{critical thresholds}. Critical thresholds define\u0000narrow bounds for the optimal solution of a problem instance such that the\u0000probability that the solution value lies outside these bounds vanishes as the\u0000instance size approaches infinity. In this paper, we extend the\u0000Erd\"{o}s-R'{e}nyi model to general hypergraphs on $n$ vertices and $M$\u0000hyperedges. We consider the problem of determining critical thresholds for the\u0000largest cardinality matching, and we show that for $M=o(1.155^n)$ the size of\u0000the maximum cardinality matching is almost surely 1. On the other hand, if\u0000$M=Theta(2^n)$ then the size of the maximum cardinality matching is\u0000$Omega(n^{frac12-gamma})$ for an arbitrary $gamma >0$. Lastly, we address\u0000the gap where $Omega(1.155^n)=M=o(2^n)$ empirically through computer\u0000simulations.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142254777","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 prove that every Hassett compactification of the moduli space of weighted stable rational curves that admits both a reduction map from the Losev-Manin compactification and a reduction map to projective space is a toric variety, whose corresponding polytope is a hypergraph associahedron (also known as a nestohedron). In addition, we present an analogous result for the moduli space of labeled weighted points in affine space up to translation and scaling. These results are interconnected, and we make their relationship explicit through the concept of ``inflation" of a hypergraph associahedron.
{"title":"Hypergraph associahedra and compactifications of moduli spaces of points","authors":"Jasper Bown, Javier González-Anaya","doi":"arxiv-2409.08611","DOIUrl":"https://doi.org/arxiv-2409.08611","url":null,"abstract":"We prove that every Hassett compactification of the moduli space of weighted\u0000stable rational curves that admits both a reduction map from the Losev-Manin\u0000compactification and a reduction map to projective space is a toric variety,\u0000whose corresponding polytope is a hypergraph associahedron (also known as a\u0000nestohedron). In addition, we present an analogous result for the moduli space\u0000of labeled weighted points in affine space up to translation and scaling. These\u0000results are interconnected, and we make their relationship explicit through the\u0000concept of ``inflation\" of a hypergraph associahedron.","PeriodicalId":501407,"journal":{"name":"arXiv - MATH - Combinatorics","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142255030","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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