Ferenc Bencs, Péter Csikvári, Guus Regts
We use Wagner's weighted subgraph counting polynomial to prove that the partition function of the anti-ferromagnetic Ising model on line graphs is real rooted and to prove that roots of the edge cover polynomial have length at most $4$. We moreover discuss how our results relate to efficient algorithms for approximately computing evaluations of these polynomials.
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We prove a theorem on how to count induced subgraphs in neighborhoods of graphs. Then we use it to prove a subgraph counting identity conjectured by McKay and Radziszowski in there work on Ramsey theory.
A refinement of Kelly's lemma for graph reconstruction for counting rooted subgraphs
本文精细化Kelly引理，研究计数根顶点与删除顶点重合的有根子图，证明一般不可能，但半径大于k时可计数固定高度k的多重集，并推广至边重构。
The goal of this short paper to advertise the method of gauge transformations (aka holographic reduction, reparametrization) that is well-known in statistical physics and computer science, but less known in combinatorics. As an application of it we give a new proof of a theorem of A. Schrijver asserting that the number of Eulerian orientations of a $d$--regular graph on $n$ vertices with even $d$ is at least $\left(\frac{\binom{d}{d/2}}{2^{d/2}}\right)^n$. We also show that a $d$--regular graph with even $d$ has always at least as many Eulerian orientations as $(d/2)$--regular subgraphs.
Survey of generalized Turán problems -- counting subgraphs
本文综述了广义Turán数$\mathrm{ex}(n,H,F)$的研究，即在$n$顶点无$F$图中最大化子图$H$副本数的极值问题，系统梳理了自Alon和Shikhelman 2016年开创性工作以来的主要进展与成果。
Parameterised subgraph counting problems are the most thoroughly studied topic in the theory of parameterised counting, and there has been significant recent progress in this area. Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of these subgraphs in some sense; here, we prove a number of hardness results for the situation in which this bounded treewidth condition does not hold, resulting in dichotomies for some special cases of the general subgraph counting problem. The p…
Counting dominating sets in a graph $G$ is closely related to the neighborhood complex of $G$. We exploit this relation to prove that the number of dominating sets $d(G)$ of a graph is determined by the number of complete bipartite subgraphs of its complement. More precisely, we state the following. Let $G$ be a simple graph of order $n$ such that its complement has exactly $a(G)$ subgraphs isomorphic to $K_{2p,2q}$ and exactly $b(G)$ subgraphs isomorphic to $K_{2p+1,2q+1}$. Then $d(G) = 2^n -1 + 2[a(G)-b(G)]$. We also show some new relations between the domination polynomial and the neighborh…
If ${\cal F}$ is a set of subgraphs $F$ of a finite graph $E$ we define a graph-counting polynomial $$ p_{\cal F}(z)=\sum_{F\in{\cal F}}z^{|F|} $$ In the present note we consider oriented graphs and discuss some cases where ${\cal F}$ consists of unbranched subgraphs $E$. We find several situations where something can be said about the location of the zeros of $p_{\cal F}$.
Counting sparse induced subgraphs in locally dense graphs
本文研究局部稠密图中稀疏诱导子图的计数，证明其数量约为 $\binom{ζn}{m}$，推广独立集计数结果并改进广义 Erdős-Rogers 函数的相关结论。
We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that the problems of counting even and odd k-vertex induced subgr…
On the number of small edge-weighted subgraphs
本文针对加权网络中小边加权子图计数的计算挑战，利用加权邻接矩阵推导了显式公式，提出了通用计算方法，为多领域研究者提供了实用分析工具。
We introduce a class of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focusses on the case in which Φdefines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φdescribes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that the…
Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. The answer is trivial for $k\leq 3$. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that for all $k\geq 4$, such graph contains $Ω(n^{1/(k-1)})$ distinct $(k-1)$-critical subgraphs. Since then no progress had been made until very recently, Hare resolved the case $k=4$ by showing that any $4$-critical graph on $n$ vertices contains at least $(8n-29)/3$ odd cycles. In this paper, we mainly focus on 4-critical graphs and develop some novel tools for counting cy…
Counting induced subgraphs with the Kromatic symmetric function
本文研究 Kromatic 对称函数的图区分能力，猜想其可区分所有图，并证明它能确定若干 4、5 顶点诱导子图及星加孤立点图的数量。
Counting subgraphs of coloring graphs
本文将色多项式推广为计数着色图中任意导出子图的$π_G^{(H)}(k)$，证明其关于颜色数$k$为多项式，研究了色对多项式并揭示了其与树度序列的等价关系，同时生成了新的图不变量。
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.
Following ideas that go back to Cannon, we show the rationality of various generating functions of growth sequences counting embeddings of convex subgraphs in locally-finite, vertex-transitive graphs with the (relative) falsification by fellow traveler property (fftp). In particular, we recover results of Cannon, of Epstein, Iano-Fletcher and Zwick, and of Calegari and Fujiwara. One of our applications concerns Schreier coset graphs of hyperbolic groups relative to quasi-convex subgroups, we show that these graphs have rational growth, the falsification by fellow traveler property, and the exi…
Fix graphs $F$ and $H$ and let $ex(n,H,F)$ denote the maximum possible number of copies of the graph $H$ in an $n$-vertex $F$-free graph. The systematic study of this function was initiated by Alon and Shikhelman [{\it J. Comb. Theory, B}. {\bf 121} (2016)]. In this paper, we give new general bounds concerning this generalized Turán function. We also determine $ex(n,P_k,K_{2,t})$ (where $P_k$ is a path on $k$ vertices) and $ex(n,C_k,K_{2,t})$ asymptotically for every $k$ and $t$. For example, it is shown that for $t \geq 2$ and $k\geq 5$ we have $ex(n,C_k,K_{2,t})=\left(\frac{1}{2k}+o(1)\right…
We present a dynamic data structure representing a graph G, which allows addition and removal of edges from G and can determine the number of appearances of a graph of a bounded size as an induced subgraph of G. The queries are answered in constant time. When the data structure is used to represent graphs from a class with bounded expansion (which includes planar graphs and more generally all proper classes closed on topological minors, as well as many other natural classes of graphs with bounded average degree), the amortized time complexity of updates is polylogarithmic.
Counting subgraphs in locally dense graphs
本文研究局部稠密图中的子图计数问题，围绕 KNRS 猜想证明粘合保持性、拟随机稳定性及近正则弱化形式，并建立其与半定优化和协正矩阵的联系。
Using the theory of negative association for measures and the notion of random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically tree-like graphs. Specifically, the corresponding free entropy density is shown to converge along any sequence of graphs whose random weak limit is a tree, and the limit is directly expressed in terms of the unique solution to a limiting cavity equation. On a Galton-Watson tree, the latter simplifies into a recursive distributional equation which can be solved ex…
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…
Counting spanning subgraphs in dense hypergraphs
提出一种简单方法估计高最小度数超图中生成子图副本数量，证明了在渐近最优最小余度界下包含特定数量Hamilton圈的条件，并简化或推广了已有结果。
Counting five-node subgraphs
本文为五顶点上21个非同构连通图推导精确计数公式，纠正五个已知错误，并通过正则图及真实网络验证，展示了新公式的有效性。
We consider parameterised subgraph-counting problems of the following form: given a graph G, how many k-tuples of its vertices have a given property? A number of such problems are known to be #W[1]-complete; here we substantially generalise some of these existing results by proving hardness for two large families of such problems. We demonstrate that it is #W[1]-hard to count the number of k-vertex subgraphs having any property where the number of distinct edge-densities of labelled subgraphs that satisfy the property is o(k^2). In the special case that the property in question depends only on…
Counting Hamiltonian paths between prescribed vertices in traceable graphs with a forbidden induced subgraph
本文研究在禁止某个诱导子图F的图中，若存在至少一条哈密顿路径连接两个指定顶点，则这两个顶点之间至少存在多少条哈密顿路径，并证明了一个尖锐二分法。
We consider the problem of counting the number of copies of a fixed graph $H$ within an input graph $G$. This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input $G$ has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that $H$ is easy if there is a linear-time algorithm for counting the number of copies of $H$ in an inp…
We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, including the case of constrained degrees. Our approach relies heavily on analytic combinatorics and on the notion of patchwork to describe the possible overlapping of copies. This paper is a version, extended to include proofs, of the paper with the same title to be presented at the Eurocomb 2017 meeting.
Counting subgraphs of coloring graphs using shadow graphs
本文提出利用影子图色多项式显式构造着色图中诱导子图计数多项式的方法，给出超立方体情形公式，并以同色对多项式、不同色多项式的图对反驳相关猜想。
We introduce a new graph invariant Λ(G) that we call maxmaxflow, and put it in the context of some other well-known graph invariants, notably maximum degree and its relatives. We prove the equivalence of two "dual" definitions of maxmaxflow: one in terms of flows, the other in terms of cocycle bases. We then show how to bound the total number (or more generally, total weight) of various classes of subgraphs of G in terms of either maximum degree or maxmaxflow. Our results are motivated by a conjecture that the modulus of the roots of the chromatic polynomial of G can be bounded above by a func…
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…
Counting independent sets in percolated graphs via the Ising model
研究通过伊辛模型在正则双部图中计数渗流子图的独立集数量，结合图容器与统计物理的簇展开方法，扩展了Kronenberg和Spinka关于渗流超立方体的工作。
Counting color-critical subgraphs under Nikiforov's condition
本文在Nikiforov条件下，推广并加强了对任意色数至少为四的色临界图F的计数结果，证明了F在G中的数量满足N_F(G)≥(γ_F-o(1))m^{(|F|-2)/2}，并建立了边谱超饱和的稳定性结果。
The chromatic polynomial $P(G,x)$ of a graph $G$ of order $n$ can be expressed as $\sum\limits_{i=1}^n(-1)^{n-i}a_{i}x^i$, where $a_i$ is interpreted as the number of broken-cycle free spanning subgraphs of $G$ with exactly $i$ components. The parameter $ε(G)=\sum\limits_{i=1}^n (n-i)a_i/\sum\limits_{i=1}^n a_i$ is the mean size of a broken-cycle-free spanning subgraph of $G$. In this article, we confirm and strengthen a conjecture proposed by Lundow and Markström in 2006 that $ε(T_n)< ε(G)<ε(K_n)$ holds for any connected graph $G$ of order $n$ which is neither the complete graph $K_n$ nor a t…
We study the problem #IndSub(P) of counting all induced subgraphs of size k in a graph G that satisfy the property P. This problem was introduced by Jerrum and Meeks and shown to be #W[1]-hard when parameterized by k for some families of properties P including, among others, connectivity [JCSS 15] and even- or oddness of the number of edges [Combinatorica 17]. Very recently [IPEC 18], two of the authors introduced a novel technique for the complexity analysis of #IndSub(P), inspired by the "topological approach to evasiveness" of Kahn, Saks and Sturtevant [FOCS 83] and the framework of graph m…
Counting cocircuits and convex two-colourings is #P-complete
证明了最多只有两种颜色的图的顶点的着色数的计数问题是P-完全的。我们还表明，密切相关的问题，计数的数量上电路的一个图是P-完全的。
Signatures of graphs for bicommutative Hopf algebras
本文利用图上的组合 Hopf 代数上的签名型泛函研究子图计数，通过特征性质和 Chen 恒等式统一代数恒等式，并证明不同子图概念对应的 Hopf 代数均同构于多项式 Hopf 代数，且同构映射保持计数运算。
We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of \emph{patchworks} to describe the possible overlappings of copies of subgraphs. Furthermore, the proofs are based on analytic combinatorics to carry out asymptotic computations. The flexibility of our approach allows us to tackle a wide range of problems. We obtain the asymptotic number and the limiting distribution of the number of subgraphs which are isomorphic to a graph from a given set of graphs. The…
In graph theory, the Szemerédi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that guarantees many copies of a subgraph $H$ provided a copy of $H$ appears in the structured component, is used in many applications to extremal problems. An analogous decomposition theorem exists for functions over $\mathbb{F}_p^n$. Specializing to $p=2$, we obtain a statement about the indicator functions of simple binary matroids. In this paper we extend previous r…
Counting induced subgraphs with given intersection sizes
本文研究诱导子图拷贝在受限交集条件下的最大计数问题，证明当交集大小集合与阶数不构成等差数列时最大数量为次多项式阶，并给出圈图及广义Turán问题的渐近结果。
We present a general method for counting and packing Hamilton cycles in dense graphs and oriented graphs, based on permanent estimates. We utilize this approach to prove several extremal results. In particular, we show that every nearly $cn$-regular oriented graph on $n$ vertices with $c>3/8$ contains $(cn/e)^n(1+o(1))^n$ directed Hamilton cycles. This is an extension of a result of Cuckler, who settled an old conjecture of Thomassen about the number of Hamilton cycles in regular tournaments. We also prove that every graph $G$ on $n$ vertices of minimum degree at least $(1/2+\varepsilon)n$ con…
Counting subgraphs in bounded-size Achlioptas processes
本文通过静态方法分析Achlioptas过程中的子图计数，推导出临界点附近最大非巨组件大小的精确界及波动分布。
Forbidden subgraphs in divisor graphs and an Erdős divisibility problem
本文研究了除数图中禁止子图的问题，证明了最大无元素相互整除的子集大小为c₂n+o(n)，并给出了此类子集数量的渐进行为。
Weighted Counting Formula and $2n/21$ Lower Bound for Induced Subgraphs with Prescribed Degree Parities
研究有限简单图中具有规定度奇偶性的诱导子图，通过定义加权计数多项式并证明等式，结合相关估计和优化得出\\(f_{\rm oe}(G)>c_*n>\frac{2n}{21}\)，改进了已有关于诱导子图阶数下界的结果。
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