Tim Silverman
We state and prove some counting formulas relating to cliques in the distant graphs of projective lines over finite rings. As a preliminary to this, we prove a decomposition theorem for the graphs in terms of the direct-product decomposition of their rings.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.
We show that the expected number of cliques in the Erdős-Rényi random graph $G(n,p)$ is $n^{\frac1{-2\log p}(\log n-2\log\log n+O(1))}$.
We consider the problem of counting $k$-cliques in $s$-uniform Erdos-Renyi hypergraphs $G(n,c,s)$ with edge density $c$, and show that its fine-grained average-case complexity can be based on its worst-case complexity. We prove the following: 1. Dense Erdos-Renyi graphs and hypergraphs: Counting $k$-cliques on $G(n,c,s)$ with $k$ and $c$ constant matches its worst-case time complexity up to a $\mathrm{polylog}(n)$ factor. Assuming randomized ETH, it takes $n^{Ω(k)}$ time to count $k$-cliques in $G(n,c,s)$ if $k$ and $c$ are constant. 2. Sparse Erdos-Renyi graphs and hypergraphs: When $c = Θ(n^…
Given a set A in Z/NZ we may form a Cayley sum graph G_A on vertex set Z/NZ by joining i to j if and only if i + j is in A. We investigate the extent to which performing this construction with a random set A simulates the generation of a random graph, proving that the clique number of G_A is a.s. O(log N). This shows that Cayley sum graphs can furnish good examples of Ramsey graphs. To prove this result we must study the specific structure of set addition on Z/NZ. Indeed, we also show that the clique number of a random Cayley sum graph on (Z/2Z)^n, 2^n = N, is almost surely not O(log N). Despi…
Counting cliques in graphs with small independence number
本文针对小独立数图的团计数问题，证明了 $k\geq 4$ 时的下界，结合已有结果说明该界对 $k=2,3,4,5$ 均最优，还得到其他超饱和结果。
The ordinary generating function of the number of complete subgraphs (cliques) of $G$, denoted by $C(G,x)$, is called the The clique polynomial of the graph $G$. In this paper, we first introduce some \emph{clique} incidence matrices associated by a simple graph $G$ as a generalization of the classical vertex-edge incidence matrix of $G$. Then, using these clique incidence matrices, we obtain two clique-counting identities that can be used for deriving two combinatorial formulas for the first and the second derivatives of clique polynomials. Finally, we conclude the paper with several open que…
Counting $t$-wise $L$-intersecting cliques with prescribed intersection sizes
本文研究 $t$-wise $L$-相交 $r$-团的最大数目，证明非等差情形下为 $o(n^{|L|})$，等差情形下给出渐近公式，推广了 Helliar 和 Liu 的结果。
Linear extensions and directed clique counts via modular partitions
本文提出通过模分区结构递归计算偏序集线性扩展的新方法，并探讨其在有向图和神经回路中的应用。
Counting cliques without generalized theta graphs
本文研究了广义theta图的广义Turán问题，确定了ex(n,K_3,H)的大小，并扩展了关于ex(n,K_r,kF)的结果。
Counting equitable $k$-colorings in graphs of bounded clique-width
研究有界团宽图中公平 $k$ 着色的计数问题，给出多项式时间算法并证明团宽依赖最优，同时改进线性团宽算法，并证明 $P_t$-free 图中公平列表 $3$ 着色可次指数时间计数。
A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. I…
Counting cliques with prescribed intersection sizes
研究限制交集大小的团广义Turán问题，改进Helliar和Liu的结果，证明当且仅当交集大小构成等差数列时渐近值成立，并给出Hilton-Milner型稳定性结果。
We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an expository treatment of our recently developed method of Randomised Algebraic Construction: we give a simpler proof of a special case of our result on clique decompositions of hypergraphs, namely triangle decompositions of quasirandom graphs.
In 1987, Kolaitis, Prömel and Rothschild proved that, for every fixed $r \in \mathbb{N}$, almost every $n$-vertex $K_{r+1}$-free graph is $r$-partite. In this paper we extend this result to all functions $r = r(n)$ with $r \leqslant (\log n)^{1/4}$. The proof combines a new (close to sharp) supersaturation version of the Erdős-Simonovits stability theorem, the hypergraph container method, and a counting technique developed by Balogh, Bollobás and Simonovits.
Counting large cliques in graphs with a forbidden tree
研究在有禁用树的图中计算大团数量的问题，当$r = k - 2$或$r = k - 3\geq5$时，验证Gerbner和Palmer的猜想，证明${\rm ex}(n,K_r,T)=a\binom{k - 1}{r}+\binom{b}{r}$并刻画极值图。
Weighted Clique and Independent Set in Edge-Distant Hereditary Graphs
本文研究遗传图类的边顶点类和边添加类上的加权最大团问题与加权最大独立集问题，并引入图到遗传类的边距离概念，证明当问题在遗传类中可多项式求解时，可在 O*(2^k) 时间内解决距离为 k 的图上的这两个问题。
Efficient Enumeration of Cliques in Graphs with Bounded Maximum Degree
本文提出了一种高效方法用于枚举具有有限最大度数的图中的 clique，并通过新证明改进了 Cutler-Radcliffe 定理和 Kahn-Zhao 定理。
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain …
We prove that counting copies of any graph $F$ in another graph $G$ can be achieved using basic matrix operations on the adjacency matrix of $G$. Moreover, the resulting algorithm is competitive for medium-sized $F$: our algorithm recovers the best known complexity for rooted 6-clique counting and improves on the best known for 9-cycle counting. Underpinning our proofs is the new result that, for a general class of graph operators, matrix operations are homomorphisms for operations on rooted graphs.
On the $4$-clique cover number of graphs
本研究针对t≥4时t-团覆盖数由Turán图T_{n,t}最大化的猜想，结合归纳框架、贪心划分等新技术，证实了t=4时的猜想，推进了团覆盖数极值问题的研究。
Solution to a problem of Katona on counting cliques of weighted graphs
针对卡托纳关于加权图团计数最小化的问题，本文确定了反例图类，并推广斯佩纳定理，证明斯佩纳图、完全多部图及弦图在最大独立集上均匀加权时团数最小。
EPG graphs, introduced by Golumbic et al. in 2009, are edge-intersection graphs of paths on an orthogonal grid. The class $B_k$-EPG is the subclass of EPG graphs where the path on the grid associated to each vertex has at most $k$ bends. Epstein et al. showed in 2013 that computing a maximum clique in $B_1$-EPG graphs is polynomial. As remarked in [Heldt et al., 2014], when the number of bends is at least $4$, the class contains $2$-interval graphs for which computing a maximum clique is an NP-hard problem. The complexity status of the Maximum Clique problem remains open for $B_2$ and $B_3$-EP…
Finding large cliques or cliques missing a few edges is a fundamental algorithmic task in the study of real-world graphs, with applications in community detection, pattern recognition, and clustering. A number of effective backtracking-based heuristics for these problems have emerged from recent empirical work in social network analysis. Given the NP-hardness of variants of clique counting, these results raise a challenge for beyond worst-case analysis of these problems. Inspired by the triadic closure of real-world graphs, Fox et al. (SICOMP 2020) introduced the notion of $c$-closed graphs an…
A Polynomial-Time Algorithm for Coloring Perfect Graphs Based on Walk Counting
该研究提出基于图论操作的多项式时间算法，核心是通过迭代计算加权路径判定完美图是否含给定大小团，权重先统一初始化再依上次迭代路径计数更新，用于对完美图进行最优着色。
Counting the minimum number of arcs in an oriented graph having weak diameter 2
本研究针对弱直径为2的n顶点有向图最小弧数f_2(n)的求解难题，通过关联绝对有向团证明其严格递增，改进了已知上界并猜想其为紧界，同时优化了有向消色数的相关界。
Spectral Erdős--Gallai Theorems for the \(\mathcal A_α\)-Tensor of the \(s\)-Clique Hypergraph
本文研究 $s$-团超图的 $\mathcal A_\alpha$-张量谱半径，确定了无 $t$ 匹配的图中 $\alpha$-$s$-团谱半径的最大值，并推广了已知结果，覆盖了 $\alpha=0$ 和 $\alpha=1$ 的情形。
Clique supersaturation under a chromatic constraint below the Turán threshold
本文针对Turán阈值以下非p部图的场景，证明了满足边数条件的图中K_{p+1}副本数的精确团计数界，其构造达到该界，为Brouwer阈值提供了团计数类似物。
The clique chromatic number of sparse random graphs
本文针对稀疏随机图 G_{n,p}，在 n^{-2/5} ≪ p ≪ 1 的大部分边概率范围内确定了团色数的数量级，通过结合概率方法与新的度数计数论证克服高度顶点带来的困难，并发现了与先前预测相矛盾的新现象。
In the counting Graph Homomorphism problem (#GraphHom) the question is: Given graphs G,H, find the number of homomorphisms from G to H. This problem is generally #P-complete, moreover, Cygan et al. proved that unless the ETH is false there is no algorithm that solves this problem in time O(|V(H)|^{o(|V(G)|)}. This, however, does not rule out the possibility that faster algorithms exist for restricted problems of this kind. Wahlstrom proved that #GraphHom can be solved in plain exponential time, that is, in time k^{|V(G)|+V(H)|}\poly(|V(H)|,|V(G)|) provided H has clique width k. We generalize t…
For any connected multigraph $G=(V,E)$ and any $M\subseteq E$, if $M$ induces an acyclic subgraph of $G$ and removing all edges in $M$ yields a subgraph of $G$ whose components are complete graphs, a formula for $τ_G(M)$ is obtained, where $τ_G(M)$ is the number of spanning trees in $G$ which contain all edges in $M$. Applying this result, we can easily obtain a formula for the number of spanning trees in the line graph or the middle graph of an arbitrary graph. Applying this result, we also show that for any connected graph $G$ with a clique $U$ which is a cut-set of $G$, the number of spanni…
On the extrema of the mean subtree order of graphs
本文证明了在阶为 $n$ 的连通图中，平均子树阶的最小值由路径取得，最大值由团取得，分别通过扩展已有思想和双重计数论证完成。
Exact Clique Counts in Graphs with a Forbidden Tree
本文针对Erdős–Sós问题，证明Gerbner和Palmer关于树T_t-自由图中r-团最大数量的猜想，还推导团谱半径的相关不等式，覆盖叶簇与少叶两种树类型。
Independent Sets and Balanced Cycle-Linkings in 2-Connected Graphs
本研究确定了2-连通图中最大化各大小独立集数量的唯一极值图，即团阶差至多1的平衡环链，并将连通图的极值结果推广至2-连通情形。
A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs
本文建立偏立方体立方体多项式与其交叉图团多项式之间的不等式及等号的中值图充要条件，并据此将中值图研究转化为一般图多项式问题，同时反驳其单峰性猜想。
Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Wil…
For two fixed graphs $T$ and $H$ let $ex(G(n,p),T,H)$ be the random variable counting the maximum number of copies of $T$ in an $H$-free subgraph of the random graph $G(n,p)$. We show that for the case $T=K_m$ and $χ(H)> m$ the behavior of $ex(G(n,p),K_m,H)$ depends strongly on the relation between $p$ and $m_2(H)=\max_{H'\subset H, |V(H')|'\geq 3}\left\{ \frac{e(H')-1}{v(H')-2} \right\}$. When $m_2(H)> m_2(K_m)$ we prove that with high probability, depending on the value of $p$, either one can maintain almost all copies of $K_m$, or it is asymptotically best to take a $χ(H)-1$ partite subgrap…
Counting Kernels in Directed Graphs with Arbitrary Orientations
该研究针对有向图核计数与判定的NP难问题，提出了不依赖边定向的模糊圆形区间图核多项式计数与存在时构造算法，并证明补可约图核问题一般NP难但阈值图子类可线性时间求解。
A SAT-based Filtering Framework for Exact Coverings of K33 by Cliques of Order 3, 4 or 5
本文提出基于 SAT 的算法框架，用于确定 K33 边的精确覆盖所需 clique 数量，通过分层搜索管道和对称性减少等方法，得出 K33 需要至少 59 个 clique 的结论。
Given a locally injective real function g on the vertex set V of a finite simple graph G=(V,E), we prove the Poincare-Hopf formula f_G(t) = 1+t sum_{x in V} f_{S_g(x)}(t), where S_g(x) = { y in S(x), g(y) less than g(x) } and f_G(t)=1+f_0 t + ... + f_{d} t^{d+1} is the f-function encoding the f-vector of a graph G, where f_k counts the number of k-dimensional cliques, complete sub-graphs, in G. The corresponding computation of f reduces the problem recursively to n tasks of graphs of half the size. For t=-1, the parametric Poincare-Hopf formula reduces to the classical Poincare-Hopf result X(G…
There are many variations on partition functions for graph homomorphisms or colorings. The case considered here is a counting or hard constraint problem in which the range or color graph carries a free and vertex transitive Abelian group action so that the colors are identified with the elements of this group. A Fourier transform is used to obtain an expansion for the numbers of colorings with terms indexed by isthmus free subgraphs of the domain. The terms are products of a polynomial in the edge density a of the color graph and the number of colorings of the indexing subgraph of the domain i…
Sparse graph counting and Kelley-Meka bounds for binary systems
本研究扩展Kelley-Meka的结果，为二元线性形式系统定义的线性模式建立强界，提出稀疏图有效的图计数引理，并将其应用于阿贝尔Cayley和图的Turán问题，得到边数上界。
The maximum spectral radius of uniform hypergraphs whose shadow excludes a complete or complete bipartite minor
本文确定了阴影不含完全或完全二部子式的均匀超图的最大谱半径，推广了Tait及Zhai-Lin的图论结果，并给出了极值超图的唯一结构及其阶的渐近估计。
On degree powers and counting stars in $F$-free graphs
本文研究无 F 图中度幂和 e_r(G) 的极值问题，通过建立其与星 S_r 拷贝数极值 ex(n,S_r,F) 的联系，证明颜色临界边情形下的极值图结构，并给出多类非二部图及 C_4 的精确结果。
Havel--Hakimi Residues of Common-Divisor Graphs: Complete Asymptotics and Prime-Counting Structure
本文针对顶点为2到n的公约数图，确定其Havel--Hakimi残差的完整渐近展开，证明无标号图可通过单纯真孪生类确定素数计数函数，所得结果与黎曼假设等价。
Counting edge-colorings of a complete graph avoiding a rainbow $K_4$
本文研究避免彩虹 $K_4$ 的完全图边着色计数，证明对 $r\ge25$ 大多数着色使用超过五种颜色，并给出指数增长率，结合容器法与图移除引理。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。