Remco van der Hofstad, Joel Spencer
We find the asymptotic number of connected graphs with $k$ vertices and $k-1+l$ edges when $k,l$ approach infinity, reproving a result of Bender, Canfield and McKay. We use the {\em probabilistic method}, analyzing breadth-first search on the random graph $G(k,p)$ for an appropriate edge probability $p$. Central is analysis of a random walk with fixed beginning and end which is tilted to the left.
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Consider the set of all digraphs on $[N]$ with $M$ edges, whose minimum in-degree and minimum out-degree are at least $k_1$ and $k_2$ respectively. For $k:=\min\{k_1,k_2\}\ge 2$ and $M/N>\max\{k_1,k_2\}$, $M=Θ(N)$, we show that, among those digraphs, the fraction of $k$-strongly connected digraphs is $1-O\bigl(N^{-(k-1)})$. Earlier with Dan Poole we identified a sharp edge-density threshold $c^*(k_1,k_2)$ for birth of a giant $(k_1,k_2)$-core in the random digraph $D(n,m=[cn])$. Combining the claims, for $c>c^*(k_1,k_2)$ with probability $1-O\bigl(N^{-(k-1)})$ the giant $(k_1,k_2)$-core exists…
The simple connected graphs may be classified by their cycle composition (number and lengths of cycles). This work derives the counting series of the simple connected graphs that have cycles of unrestricted number and length, but no overlapping cycles. Cycle pairs of these graphs of interest must not have common nodes or edges. The recipe of counting these graphs is based on the counting series of the associated planted graphs, multisets of planted graphs, a recursive synthesis of enriched trees, and a generalized Otter's formula that maps the underlying rooted block graphs to the underlying b…
Let $G=(V(G),E(G))$ be a simple graph. A non-empty set $S\subseteq V (G)$ is a weakly connected dominating set in $G$, if the subgraph obtained from $G$ by removing all edges each joining any two vertices in $V (G)\setminus S$ is connected. In this paper, we consider some graphs and study the number of their weakly connected dominating sets.
We enumerate the connected graphs that contain a linear number of edges with respect to the number of vertices. So far, only the first term of the asymptotics was known. Using analytic combinatorics, i.e. generating function manipulations, we derive the complete asymptotic expansion.
A sharp asymptotic formula for the number of strongly connected digraphs on $n$ labelled vertices with $m$ arcs, under a condition $m-n\to\infty$, $m=O(n)$, is obtained; this solves a problem posed by Wright back in $1977$. Our formula is a counterpart of a classic asymptotic formula, due to Bender, Canfield and McKay, for the total number of connected undirected graphs on $n$ vertices with $m$ edges. A key ingredient of their proof was a recurrence equation for the connected graphs count due to Wright. No analogue of Wright's recurrence seems to exist for digraphs. In a previous paper with Ni…
In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]=\{1,2,\ldots,n\}$ with $m$ edges, whenever $n\to\infty$ and $n-1\le m=m(n)\le \binom{n}{2}$. We give an asymptotic formula for the number $C_r(n,m)$ of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges, whenever $r\ge 3$ is fixed and $m=m(n)$ with $m/n\to\infty$, i.e., the average degree tends to infinity. This complements recent results of Behrisch, Coja-Oghlan and Kang (the case $m=n/(r-1)+Θ(n)$) and the present authors (the case $m=n/(r-1)+o(n)$, i.e., `nullity' or `excess' $o(n)…
Arthur Cayley famously proved that there are n to the power n-2 labeled trees on n vertices. Here we go much further and show how to enumerate, fully automatically, labeled trees such that every vertex has a number of neighbors that belongs to a specified finite set, and also count trees where the number of neighbors is not allowed to be in a given finite set. We also give detailed statistical analysis, and show that in the sample space of labeled trees with n vertices, the random variable "number of vertices with d neighbors" is asymptotically normal, and for any different degrees, are jointl…
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…
In this paper we count the number of rows f_n with the value "false" in the truth tables of all bracketed formulae with n distinct variables connected by the binary connective of implication. We find a recurrence and an asymptotic formulae for f_n. We also show that the ratio of f_n to the total number of rows converges to \frac{3-\sqrt{3}}{6}.
In this paper we count the number of rows yn with the value "false" in the truth tables of all bracketed formulae with n distinct variables connected by the binary connective of 2modified-implication". We find a recurrence and an asymptotic formulae for yn. We also determine the parity of yn.
In this paper we asymptotically count $d$-regular $k$-uniform hypergraphs on $n$ vertices, provided $k$ is fixed and $d=d(n)=o(n^{1/2})$. In doing so, we extend to hypergraphs a switching technique of McKay and Wormald.
Counting $k$-cycles in $5$-connected planar triangulations
该研究证明了5-连通平面三角剖分中5圈数上界为9n-50且紧，并给出k≥6时k圈数的渐近紧上界C(k)·n^{⌊k/3⌋}。
Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $π=\{V_1, ..., V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $π$ is connected for $1\le j\le k$. Define $q_{i}(G)$ as the number of connected set partitions in $G$ with $i$ blocks. The partition polynomial is then $Q(G, x)=\sum_{i=0}^n q_{i}(G)x^i$. This paper presents a splitting approach to the partition polynomial on a separating vertex set $X$ in $G$ and summarizes some properties of the bond lattice. Furthermore the bivariate partition p…
Let $S_k(n)$ be the maximum number of orientations of an $n$-vertex graph $G$ in which no copy of $K_k$ is strongly connected. For all integers $n$, $k\geq 4$ where $n\geq 5$ or $k\geq 5$, we prove that $S_k(n) = 2^{t_{k-1}(n)}$, where $t_{k-1}(n)$ is the number of edges of the $n$-vertex $(k-1)$-partite Turán graph $T_{k-1}(n)$, and that $T_{k-1}(n)$ is the only $n$-vertex graph with this number of orientations. Furthermore, $S_4(4) = 40$ and this maximality is achieved only by $K_4$.
We use counting arguments to show that asymptotically almost all sparse paving matroids contain an $H$-minor, where $H$ falls into one of several simple classes of matroids. Furthermore the result holds for all $H$ in a larger class of matroids, if we restrict to asymptotically almost all sparse paving matroids of fixed rank $r$ (where $r$ is necessarily no smaller than the rank of $H$).
We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question as to whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguin…
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Ω(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Ω(n^{1/4})}$ Hamiltonian cycles.
We present a new method to count unrooted maps on the sphere up to orientation-preserving homeomorphisms. The principle, called tree-decomposition, is to deform a map into an arborescent structure whose nodes are occupied by constrained maps. Tree-decomposition turns out to be very efficient and flexible for the enumeration of constrained families of maps. In this article, the method is applied to count unrooted 2-connected maps and, more importantly, to count unrooted 3-connected maps, which correspond to the combinatorial types of oriented convex polyhedra. Our method improves significantly …
For various quadruple systems F, we give asymptotically sharp lower bounds on the number of copies of F in a quadruple system with a prescribed number of vertices and edges. Our results extend those of Furedi, Keevash, Pikhurko, Simonovits and Sudakov who proved under the same conditions that there is one copy of $F$. Our proofs use the hypergraph removal Lemma and stability results for the corresponding Turan problem proved by the above authors.
Let A_n (n >= 1) be the set of all integers x such that there exists a connected graph on n vertices with precisely x spanning trees. In this paper, we show that |A_n| grows faster than sqrt{n}exp(2Pi*sqrt{n/log{n}/Sqrt(3)} This settles a question of Sedlacek.
Let C(n,k) denote the number of connected graphs with n labeled vertices and n+k-1 edges. For any sequence (k_n), the limit of C(n,k_n) as n tends to infinity is known. It has been observed that, if k_n=o(\sqrt{n}), this limit is asymptotically equal to the $k_n$th moment of the area under the standard Brownian excursion. These moments have been computed in the literature via independent methods. In this article we show why this is true for k_n=o(\sqrt[3]{n}) starting from an observation made by Joel Spencer. The elementary argument uses a result about strong embedding of the Uniform empirical…
A family of subsets of $\{1,\ldots,n\}$ is called {\it intersecting} if any two of its sets intersect. A classical result in extremal combinatorics due to Erdős, Ko, and Rado determines the maximum size of an intersecting family of $k$-subsets of $\{1,\ldots, n\}$. In this paper we study the following problem: how many intersecting families of $k$-subsets of $\{1,\ldots, n\}$ are there? Improving a result of Balogh, Das, Delcourt, Liu, and Sharifzadeh, we determine this quantity asymptotically for $n\ge 2k+2+2\sqrt{k\log k}$ and $k\to \infty$. Moreover, under the same assumptions we also deter…
In this short note we determine the maximum number, over all $n$-vertex graphs $G$, of orientations of $G$ containing no strongly connected cycle $C_{2k+1}$. This answers a part of a recent question of Araujo, Botler and Mota.
For $G=G_{n, 1/2}$, the Erdős--Renyi random graph, let $X_n$ be the random variable representing the number of distinct partitions of $V(G)$ into sets $A_1, \ldots, A_q$ so that the degree of each vertex in $G[A_i]$ is divisible by $q$ for all $i\in[q]$. We prove that if $q\geq 3$ is odd then $X_n\xrightarrow{d}{\mathrm{Po}(1/q!)}$, and if $q \geq 4$ is even then $X_n\xrightarrow{d}{\mathrm{Po}(2^q/q!)}$. More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in $G[A_i]$ to be congruent to $x_i$ modulo $q$ for each $i\in[q]$, where the residue…
We denote a polygon as a connected component in Cayley tree of order 2 containing certain number of fix vertices. We found an exact formula for a polygon counting problem for two cases, in which, for the first case the polygon contain a full connected component of a Cayley tree and for the second case the polygon contain two fixed vertices. From these formulas, which is in the form of finite linear combination of Catalan numbers, one can find the asymptotic estimation for a counting problem.
A fundamental theorem in graph theory states that any 3-connected graph contains a subdivision of $K_4$. As a generalization, we ask for the minimum number of $K_4$-subdivisions that are contained in every $3$-connected graph on $n$ vertices. We prove that there are $Ω(n^3)$ such $K_4$-subdivisions and show that the order of this bound is tight for infinitely many graphs. We further investigate a better bound in dependence on $m$ and prove that the computational complexity of the problem of counting the exact number of $K_4$-subdivisions is $\#P$-hard.
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a si…
In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]$ with $m$ edges, whenever $n$ and the nullity $m-n+1$ tend to infinity. Asymptotic formulae for the number of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges and so nullity $t=(r-1)m-n+1$ were proved by Karoński and Łuczak for the case $t=o(\log n/\log\log n)$, and Behrisch, Coja-Oghlan and Kang for $t=Θ(n)$. Here we prove such a formula for any $r\ge 3$ fixed, and any $t=t(n)$ satisfying $t=o(n)$ and $t\to\infty$ as $n\to\infty$. This leaves open only the (much simpler) case $t/n…
We prove an asymptotically tight bound (asymptotic with respect to the number of polynomials for fixed degrees and number of variables) on the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of real polynomials. More precisely, we prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of a family of $s$ polynomials in $\R[X_1,...,X_k]$ whose degrees are at most $d$ is bounded by \[ \frac{(2d)^k}{k!}s^k + O(s^{k-1}). \] This improves the best upper bound known prev…
Counting Connected Partitions of Graphs
受Győri-Lovász定理启发，本文研究了图的连通划分计数问题，分别针对边划分和点划分情形，证明了关于可实现连通子图边数或顶点数的无序正整数解个数的多个紧致下界。
A counting version of Petersen's $2$-factor theorem
该论文证明了每个 $n$ 顶点 $2r$-正则简单图至少含有 $\bigl((1+o_r(1))\frac{2r}{e}\bigr)^n$ 个不同的 $2$-因子，改进了先前下界，并由此确定了 $2$-因子分解数量的渐近紧界。
We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of counting the number of cocircuits of a graph is #P-complete.
Counting Small Cycle Double Covers
该研究扩展了Seyffarth的小环双覆盖定理，证明平面4-连通图含线性数量的小CDC并给出构造补充，同时强化三次图CDC计数引理，更简洁地证明了平面2-连通三次图的小CDC存在性结论。
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…
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to $g$? We outline Zhai's proof of a conjecture of Bras-Amorós that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems. We highlight the many contributions made by undergraduates to problems in this area.
We present an average case analysis of two variants of dual-pivot quicksort, one with a non-algorithmic comparison-optimal partitioning strategy, the other with a closely related algorithmic strategy. For both we calculate the expected number of comparisons exactly as well as asymptotically, in particular, we provide exact expressions for the linear, logarithmic, and constant terms. An essential step is the analysis of zeros of lattice paths in a certain probability model. Along the way a combinatorial identity is proven.
We compute the number of equivalence classes of nonperiodic covering cycles of given length in a non oriented connected graph. A covering cycle is a closed path that traverses each edge of the graph at least once. A special case is the number of Euler cycles in the non oriented graph. An identity relating the numbers of covering cycles of any length in a graph to a product of determinants is obtained.
For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We show that for all $0\le \ell<k-1$, every $k$-graph with minimum co-degree $δn$ with $δ>1/2$ has (asymptotically and up to a subexponential factor) at least as many Hamiltonian $\ell$-cycles as in a typical random $k$-graph with edge-probability $δ$. This significantly improves a recent result of Glock, Gould, Joos, Kühn, and Osthus, and verifies a conjectur…
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…
In this paper we present a method for producing asymptotic estimates for the number of integers in a given S having only ``small'' prime factors. The conditions that need to be verified are simpler than those required by other methods, and we apply our result to give an easy proof of a result which says that dense subsets A and B of {1,2,...,x} always produce asymptotically the expected number of x^r - smooth sums a+b, where a in A and b in B. Recall that a number n is said to be y-smooth if all its prime divisors are at most y.
For a family X of k-subsets of the set 1,...,n, let |X| be the cardinality of X and let Gamma(X,mu) be the expected maximum weight of a subset from X when the weights of 1,...,n are chosen independently at random from a symmetric probability distribution mu on R. We consider the inverse isoperimetric problem of finding mu for which Gamma(X,mu) gives the best estimate of ln|X|. We prove that the optimal choice of mu is the logistic distribution, in which case Gamma(X,mu) provides an asymptotically tight estimate of ln|X| as k^{-1}ln|X| grows. Since in many important cases Gamma(X,mu) can be eas…
On Counting Constructions and Isomorphism Classes of $I$-Graphs
本文利用解析数论技术证明了 $I$-图的渐近密度结果，精确量化了其中作为广义 Petersen 图及连通图的比例。
Multiparameter counting of numerical semigroups: recurrences and leaf-discriminating trees
研究数值半群多参数计数问题，证明特定条件下的计数公式，提出种子算法的两种改编用于探索相关树结构，并改进算法策略，扩展对序列的认识，计算了多种情况下的半群数量及多参数分解。
The maximal number of totally mixed Nash equilibria in games of several players equals the number of block derangements, as proved by McKelvey and McLennan.On the other hand, counting the derangements is a well studied problem. The numbers are identified as linearization coefficients for Laguerre polynomials. MacMahon derived a generating function for them as an application of his master theorem. This article relates the algebraic, combinatorial and game-theoretic problems that were not connected before. New recurrence relations, hypergeometric formulas and asymptotics for the derangement coun…
A vertical 2-sum of a two-coatom lattice $L$ and a two-atom lattice $U$ is obtained by removing the top of $L$ and the bottom of $U$, and identifying the coatoms of $L$ with the atoms of $U$. This operation creates one or two nonisomorphic lattices depending on the symmetry case. Here the symmetry cases are analyzed, and a recurrence relation is presented that expresses the number of such vertical 2-sums in some family of interest, up to isomorphism. Nonisomorphic, vertically indecomposable modular and distributive lattices are counted and classified up to 35 and 60 elements respectively. Asym…
Directed acyclic graphs (DAGs) can be characterised as directed graphs whose strongly connected components are isolated vertices. Using this restriction on the strong components, we discover that when $m = cn$, where $m$ is the number of directed edges, $n$ is the number of vertices, and $c < 1$, the asymptotic probability that a random digraph is acyclic is an explicit function $p(c)$, such that $p(0) = 1$ and $p(1) = 0$. When $m = n(1 + μn^{-1/3})$, the asymptotic behaviour changes, and the probability that a digraph is acyclic becomes $n^{-1/3} C(μ)$, where $C(μ)$ is an explicit function of…
A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this, is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so called Pos?a `rotation-extension' technique for the undirected analogue. Let ${\mathcal D}(n,p)$ denote the random digraph on vertex set $[n]$, obtained by adding each directed edge independently with probability $p$. Here, we prese…
Phylogenetic networks generalize phylogenetic trees, and have been introduced in order to describe evolution in the case of transfer of genetic material between coexisting species. There are many classes of phylogenetic networks, which can all be modeled as families of graphs with labeled leaves. In this paper, we focus on rooted and unrooted level-k networks and provide enumeration formulas (exact and asymptotic) for rooted and unrooted level-1 and level-2 phylogenetic networks with a given number of leaves. We also prove that the distribution of some parameters of these networks (such as the…
Two edge colorings of a graph are {\em edge-Kempe equivalent} if one can be obtained from the other by a series of edge-Kempe switches. This work gives some results for the number of edge-Kempe equivalence classes for cubic graphs. In particular we show every 2-connected planar bipartite cubic graph has exactly one edge-Kempe equivalence class. Additionally, we exhibit infinite families of nonplanar bipartite cubic graphs with a range of numbers of edge-Kempe equivalence classes. Techniques are developed that will be useful for analyzing other classes of graphs as well.
The counting of alternating tangles in terms of their crossing number, number of external legs and connected components is presented here in a unified framework using quantum field-theoretic methods applied to a matrix model of colored links. The overcounting related to topological equivalence of diagrams is removed by means of a renormalization scheme of the matrix model; the corresponding ``renormalization equations'' are derived. Some particular cases are studied in detail and solved exactly.
Random Turán and counting results for general position sets over finite fields
本文研究有限域随机子集中一般位置集的最大规模与计数，利用超图容器及二维点线关联图的伪随机性，给出二维数量级、高维近紧上界和渐近紧计数界。
A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number of matroids on $n$ elements whose cover complexity is bounded. We apply cover complexity to show that the class of matroids without an $N$-minor is asymptotically small in case $N$ is one of the sparse paving matroids $U_{2,k}$, $U_{3,6}$, $P_6$, $Q_6$, or $R_6$, thus confirming a few special cases of a conjecture due to Mayhew, Newman, Welsh, and Whittle. …
In 2006, Barat and Thomassen conjectured in 2006 that the edges of every planar 4-regular 4-edge-connected graph can be decomposed into copies of the star with 3 leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Using the small subgraph conditioning method of Robinson and Wormald, we prove that a random 4-regular graph has an $S_3$-decomposition asymptotically almost surely, provided the number of vertices is divisible by 3.
We define numbers of the type Oj(N) and Ej(N) and the corresponding integer sequences. We prove that these integer sequences, e.g., SO(N) and SE(N) correspond to the number of odd and even walks in complete graphs. We then prove that there is a unique family of graphs which have exactly the same sequence of odd walks between connected nodes and of even walks between pairs of nodes at distance two, respectively. These graphs are obtained as the Kronecker product. We show that they are the incidence graphs of block designs, are distance-regular and Ramanujan graphs.
In 1930, Kuratowski showed that $K_{3,3}$ and $K_5$ are the only two minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň and Kochol showed that there are infinitely many $k$-crossing-critical graphs for any $k\ge 2$, even if restricted to simple $3$-connected graphs. Recently, $2$-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such gr…
In this note we asymptotically determine the maximum number of hyperedges possible in an $r$-uniform, connected $n$-vertex hypergraph without a Berge path of length $k$, as $n$ and $k$ tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity.
We consider the following question of Bollobas: given an r-colouring of the edges of the complete graph on n vertices, how large a k-connected subgraph can we find using only one colour? We solve this problem asymptotically when r-1 is a prime power, and exactly for 2- and 3-colourings.
An Asymptotically Tight $t\log t$ Bound for $k$-Connected Subgraphs in Dense $K_t$-Minor-Free Graphs
研究在稠密无 \(K_t\) 子图的图中 \(k\) - 连通子图的界，通过改进方法，将结构界从 \(O(t\log^3 t)\) 提升到 \(O(t\log^2 t)\)，优化了相关图的阶，从 \(O(t\log^4 t)\) 变为 \(O(t\log^3 t)\)。
Counting paths in directed graphs
研究有向图中长度为k的路径数量最大化的图结构，并探讨其在半环中的推广问题
We investigate the asymptotic number of elements of size $n$ in a particular class of closed lambda-terms (so-called $BCI(p)$-terms) which are related to axiom systems of combinatory logic. By deriving a differential equation for the generating function of the counting sequence we obtain a recurrence relation which can be solved asymptotically. We derive differential equations for the generating functions of the counting sequences of other more general classes of terms as well: the class of $BCK(p)$-terms and that of closed lambda-terms. Using elementary arguments we obtain upper and loweresti…
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