Hunter S. Snevily, Douglas B. West
We introduce a combinatorial enumeration problem that is solved using generalized Catalan numbers. We also study generalizations of the Cycle Lemma beyond the computation of the generalized Catalan numbers.
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We correct some errors and omissions primarily in a paper [Albertson&Hutchinson2004], discovered by R.B. Richter, and also some in a proof of [Thomassen1993] and of [Yu1997]. We give a short proof of Thomassen's theorem that every triangulation of a surface with all noncontractible cycles sufficiently long can be 5-colored; part of the shortness is due to the use of the Four Color Theorem, which is not used in Thomassen's original proof.
In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph $G_{n,p}$ for $p \geq C(\log n/n)^{1/Δ}$, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph $H$ with $Δ(H) \leq Δ$. However, this is typically only optimal when $Δ\in \{2,3\}$ and $H$ either contains a triangle ($Δ= 2$) or many copi…
We present a non-commutative version of the cycle lemma of Dvoretsky and Motzkin that applies to free groups and use this result to solve a number of problems involving cyclic reduction in the free group. We also describe an application to random matrices, in particular the fluctuations of Kesten's Law.
In this paper we give a proof of Enomoto's conjecture for graphs of sufficiently large order. Enomoto's conjecture states that, if $G$ is a graph of order $n$ with minimum degree $δ(G)\geq \frac{n}{2}+1$, then for any pair of vertices $x$, $y$ in $G$, there is a Hamiltonian cycle $C$ of $G$ such that $d_C(x,y)=\lfloor \frac{n}{2}\rfloor$. The main tools of our proof are Regularity Lemma of Szemerédi and Blow-up Lemma of Komlós et al.
In this paper, we develop a method for studying cycle lengths in hypergraphs. Our method is built on earlier ones used in [21,22,18]. However, instead of utilizing the well-known lemma of Bondy and Simonovits [4] that most existing methods do, we develop a new and very simple lemma in its place. One useful feature of the new lemma is its adaptiveness for the hypergraph setting. Using this new method, we prove a conjecture of Verstraëte [37] that for $r\ge 3$, every $r$-uniform hypergraph with average degree $Ω(k^{r-1})$ contains Berge cycles of $k$ consecutive lengths. This is sharp up to the …
For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for any abelian group $G$. The best known bounds relating the parameters in the lemma for general $G$ are of tower-type. For $k>3$, even in the case $G=\mathbb{F}_2^n$ no better bounds were known prior to this paper. This special case has received considerable attention due to its close connection to property testing of boolean functions. For every $k\geq 3$, we prove a polynomial bound relating the parameters for $G=\mathbb{F}_p^n$, where $p$ is any fixed prime. This extends the result for $k=3$ by the first two authors. D…
We give a new proof of the Skeletal Lemma, which is the main technical tool in our paper on Hamilton cycles in line graphs [T. Kaiser and P. Vrána, Hamilton cycles in 5-connected line graphs, European J. Combin. 33 (2012), 924-947]. It generalises results on disjoint spanning trees in graphs to the context of 3-hypergraphs. The lemma is proved in a slightly stronger version that is more suitable for applications. The proof is simplified and formulated in a more accessible way.
Sachs and Conway-Gordon used linking number and a beautiful counting argument to prove that every graph in the Petersen family is intrinsically linked (have a pair of disjoint cycles that form a nonsplit link in every spatial embedding) and thus each family member has no flat spatial embedding (an embedding for which every cycle bounds a disk with interior disjoint from the graph). We give an alternate proof that every Petersen family graph has no flat embedding by applying Böhme's Lemma and the Jordan-Brouwer Separation Theorem.
The method of hypergraph containers, introduced recently by Balogh, Morris, and Samotij, and independently by Saxton and Thomason, has proved to be an extremely useful tool in the study of various monotone graph properties. In particular, a fairly straightforward application of this technique allows one to locate, for each non-bipartite graph $H$, the threshold at which the distribution of edges in a typical $H$-free graph with a given number of edges undergoes a transition from 'random-like' to 'structured'. On the other hand, for non-monotone hereditary graph properties the standard version …
Ramsey--Turán Factors of Non-directed Oriented Cycles in Oriented Graphs
该研究针对有向图，证明满足最小半度阈值及独立数条件的足够大的有向图包含指定长度的反有向环因子，其最小半度阈值渐近紧，采用了Ramsey-Turán型格吸收引理完成证明。
Motivated by a question on the maximal number of vertex disjoint Schrijver graphs in the Kneser graph, we investigate the following function, denoted by $f(n,k)$: the maximal number of Hamiltonian cycles on an $n$ element set, such that no two cycles share a common independent set of size more than $k$. We shall mainly be interested in the behavior of $f(n,k)$ when $k$ is a linear function of $n$, namely $k=cn$. We show a threshold phenomenon: there exists a constant $c_t$ such that for $c<c_t$, $f(n,cn)$ is bounded by a constant depending only on $c$ and not on $n$, and for $c_t <c$, $f(n,cn)…
A cycle $C$ of a graph $G$ is \emph{isolating} if every component of $G-V(C)$ is a single vertex. We show that isolating cycles in polyhedral graphs can be extended to larger ones: every isolating cycle $C$ of length $6 \leq |E(C)| < \left \lfloor \frac{2}{3}(|V(G)|+4) \right \rfloor$ implies an isolating cycle $C'$ of larger length that contains $V(C)$. By "hopping" iteratively to such larger cycles, we obtain a powerful and very general inductive motor for proving long cycles and computing them (we will give an algorithm with quadratic running time). This is the first step towards the so far…
The sandpile model on the complete split graph: $q,t$-Schröder polynomials, sawtooth polyominoes, and a cycle lemma
本文研究了完全分裂图上沙堆模型的有序复发配置，通过引入两种塌陷过程和相关统计量，建立了与Schroder路径的联系，并提出了锯齿多米诺的概念。
For given graphs $G_1,\ldots,G_k$, the size-Ramsey number $\hat{R}(G_1,\ldots,G_k)$ is the smallest integer $m$ for which there exists a graph $H$ on $m$ edges such that in every $k$-edge coloring of $H$ with colors $1,\ldots,k$, $ H $ contains a monochromatic copy of $G_i$ of color $i$ for some $1\leq i\leq k$. We denote $\hat{R}(G_1,\ldots,G_k)$ by $\hat{R}_{k}(G)$ when $G_1=\cdots=G_k=G$. Haxell, Kohayakawa and Łuczak showed that the size Ramsey number of a cycle $C_n$ is linear in $n$ i.e. $\hat{R}_{k}(C_{n})\leq c_k n$ for some constant $c_k$. Their proof, is based on the regularity lemma…
Sharp threshold for Hamilton cycles in randomly perturbed sparse graphs
研究随机扰动稀疏图中哈密顿圈存在的精确概率阈值，通过改进常数至最优值1并证明下界最优性。
The investigation of conditions guaranteeing the appearance of cycles of certain lengths is one of the most well-studied topics in graph theory. In this paper we consider a problem of this type which asks, for fixed integers ${\ell}$ and $k$, how many copies of the $k$-cycle guarantee the appearance of an $\ell$-cycle? Extending previous results of Bollobás--Győri--Li and Alon--Shikhelman, we fully resolve this problem by giving tight (or nearly tight) bounds for all values of $\ell$ and $k$. We also present a somewhat surprising application of the above mentioned estimates to the study of the…
Improved bound on the number of cycle sets
本文研究具有 n 个顶点的图的不同圈集数量，将 Verstraëte 的 $2^{n - n^{1/10}}$ 上界改进为 $2^{n - n^{1/2 - o(1)}}$，核心贡献是建立了针对哈密顿图的近最优容器引理。
In 1962 Pósa conjectured that every graph G on n vertices with minimum degree at least 2n/3 contains the square of a hamiltonian cycle. In 1996 Fan and Kierstead proved the path version of Pósa's Conjecture. They also proved that it would suffice to show that G contains the square of a cycle of length greater than 2n/3. Still in 1996, Komlós, Sárközy, and Szemerédi proved Pósa's Conjecture, using the Regularity and Blow-up Lemmas, for graphs of order n > n_0, where n_0 is a very large constant. Here we show without using these lemmas that n_0=2\times 10^8 is sufficient. We are motivated by the…
A remark on continued fractions for permutations and D-permutations with a weight $-1$ per cycle
该研究证明了每个循环带权重-1时，带多独立统计量的排列与D-排列的普通生成函数存在简洁连分数形式，并借助关联模2循环数与不动点、循环峰谷、交叉数的引理完成证明。
Let $k,l,m$ be integers and $r(k,l,m)$ be the minimum integer $N$ such that for any red-blue-green coloring of $K_{N,N}$, there is a red matching of size at least $k$ in a component, or a blue matching of at least size $l$ in a component, or a green matching of size at least $m$ in a component. In this paper, we determine the exact value of $r(k,l,m)$ completely. Applying a technique originated by Łuczak that applies Szemerédi's Regularity Lemma to reduce the problem of showing the existence of a monochromatic cycle to show the existence of a monochromatic matching in a component, we obtain th…
A graph-theoretic parameter, in a form of a function, called the extra-factorial sum is discussed. The main results are presented in ref. [1] (Nastou et al., Optim Lett, 10, 1203-1220, 2016) and the reader is strongly advised to study the aforementioned paper. The current work presents subject matter in a tutorial form with proofs and some newer unpublished results towards the end (lemma six extension and lemma seven). The extra-factorial sum is relevant to Hamiltonian cycles of complete weighted graphs $WH_n$ with $n$ vertices and is obtained for each edge of $WH_n$. If this sum is multiplied…
For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$ In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results. I…
Given a positive integer $ r $, the $ r $-color size-Ramsey number of a graph $ H $, denoted by $ \hat{R}(H, r) $, is the smallest integer $ m $ for which there exists a graph $ G $ with $ m $ edges such that, in any edge coloring of $ G $ with $ r $ colors, $G$ contains a monochromatic copy of $ H $. Haxell, Kohayakawa and Łuczak showed that the size-Ramsey number of a cycle $ C_n $ is linear in $ n $ i.e. $ \hat{R}(C_n, r) \leq c_rn $, for some constant $ c_r $. Their proof, however, is based on the Szemerédi's regularity lemma and so no specific constant $ c_r $ is known. Javadi, Khoeini, O…
The extremal number of cycles with all diagonals
本文解决了Erdős提出的关于无所有对角线环的图的最大边数问题，证明了极值数为Θ(n^{3/2})，并引入了一个关于几乎跨越鲁棒扩张器的新引理。
The Connected Bipartite Turán Problem for Long Cycles and Paths
本文针对Caro等人提出的连通二部图极值边数问题，精确确定了不含特定长度路径的连通二部图的极值边数与结构，还建立了不含长环的2-连通二部图的精确结构，并应用结果简洁证明了经典定理。
Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. Rödl, A. %Ruciński, M. Simonovits, J. Skokan, The Ramsey number for hypergraph cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color Ramsey number of 3-uniform loose cycles on $2n$ vertices is asymptotically $\frac{5n}{2}$. Their proof is based on the method of Regularity Lemma. Here, without using this method, we generalize their result by determining the exact values of 2-color Ramsey numbers involving loose paths and cycles in 3-uniform hypergraphs. More precisely, we prove that for every $n\geq m\geq 3$, $R(\mathc…
In this paper, we study the relations between the numerical structure of the optimal solutions of a convex programming problem defined on the edge set of a simple graph and the stability number (i.e. the maximum size of a subset of pairwise non-adjacent vertices) of the graph. Our analysis shows that the stability number of every graph G can be decomposed in the sum of the stability number of a subgraph containing a perfect 2-matching (i.e. a system of vertex-disjoint odd-cycles and edges covering the vertex-set) plus a term computable in polynomial time. As a consequence, it is possible to bo…
A classic result of Erdős and, independently, of Bondy and Simonovits says that the maximum number of edges in an $n$-vertex graph not containing $C_{2k}$, the cycle of length $2k$, is $O( n^{1+1/k})$. Simonovits established a corresponding supersaturation result for $C_{2k}$'s, showing that there exist positive constants $C,c$ depending only on $k$ such that every $n$-vertex graph $G$ with $e(G)\geq Cn^{1+1/k}$ contains at least $c\left(\frac{e(G)}{v(G)}\right)^{2k}$ many copies of $C_{2k}$, this number of copies tightly achieved by the random graph (up to a multiplicative constant). In this …
Imagine that you are handed a rule for determining whether a cycle in a digraph is "good" or "bad", based on which edges of the cycle are traversed in the forward direction and which edges are traversed in the backward direction. Can you then construct a digraph which avoids having any "bad" cycles, but has arbitrarily large chromatic number? We answer this question when the rule is described in terms of a finite state machine. The proof relies on Nesetril and Rodl's structural Ramsey theory of posets with a linear extension. As an application, we give a new proof of the Loop Lemma of Barto, K…
A sharp asymptotic bound for odd cycles in planar graphs
该研究针对平面图中奇环的最大副本数问题，证明了一个精确渐近界公式，解决了Heath等人提出的相关加权优化猜想，确定了所有固定奇环的主项。
Dichromatic Number and Cycle Inversions
本文用Farkas引理简洁证明了有向图无环色数的循环排序刻画，并表明通过逐次反转一族有向循环可将其降至至多2。
Edge-disjoint Hamilton cycles under a bipartite-hole condition
研究在二分洞条件下，图中边不相交哈密顿圈的存在情况。通过引入函数\(f(a,k)\)，利用删除引理、哈密顿性定理及贪心填充论证证明其上下界，即\(f(a,k)=\Theta\left(a+k+\frac{ak}{\log(k+2)}\right)\)。
(Dieudonné and) Dwork's lemma gives a necessary and sufficient condition for an exponential of a formal power series $S(z)$ with coefficients in $Q_p$ to have coefficients in $Z_p$. We establish theorems on the $p$-adic valuation of the coefficients of the exponential of $S(z)$, assuming weaker conditions on the coefficients of $S(z)$ than in Dwork's lemma. As applications, we provide several results concerning lower bounds on the $p$-adic valuation of the number of permutation representations of finitely generated groups. In particular, we give fairly tight lower bounds in the case of an arbi…
Let $C_{n}$ be a cycle of length $n$. As an application of Szemerédi's regularity lemma, Łuczak ($R(C_n,C_n,C_n)\leq (4+o(1))n$, J. Combin. Theory Ser. B, 75 (1999), 174--187) in fact established that $K_{(8+o(1))n}\to(C_{2n+1},C_{2n+1},C_{2n+1})$. In this paper, we strengthen several results involving cycles. Let $\mathcal{G}(n,p)$ be the random graph. We prove that for fixed $0<p\le1$, and integers $n_1$, $n_2$ and $n_3$ with $n_1\ge n_2\ge n_3$, it holds that for any sufficiently small $δ>0$, there exists an integer $n_0$ such that for all integer $n_3>n_0$, we have a.a.s. that \begin{align…
The inverse nullity pair problem and the strong nullity interlacing property
本文研究了图论中的逆 $i$-零度对问题，针对完全图、环和树探讨了矩阵零度对的实现，并引入强零度交错性质及超图引理和反收缩引理作为构造矩阵的新工具。
For positive integers $n>d\geq k$, let $ϕ(n,d,k)$ denote the least integer $ϕ$ such that every $n$-vertex graph with at least $ϕ$ vertices of degree at least $d$ contains a path on $k+1$ vertices. Many years ago, Erdős, Faudree, Schelp and Simonovits proposed the study of the function $ϕ(n,d,k)$, and conjectured that for any positive integers $n>d\geq k$, it holds that $ϕ(n,d,k)\leq \lfloor\frac{k-1}{2}\rfloor\lfloor\frac{n}{d+1}\rfloor+ε$, where $ε=1$ if $k$ is odd and $ε=2$ otherwise. In this paper we determine the values of the function $ϕ(n,d,k)$ exactly. This confirms the above conjecture…
For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erdős and Simonovits \cite{ESi1} conjectured that for every family $\mathcal{F}$ of bipartite graphs, there exists $k$ such that $\ex{n}{\mathcal{F} \cup \mathcal{C}_k} \sim \ex{n}{\mathcal{F} \cup \mathcal{C}}$ as $n \rightarrow \infty$. This conjecture was proved by Erdős and Simonovits when $\mathcal{F} = \{C_4\}$, and for certain families of even cycles in \cite{KSV}. In this paper, we give a general approach …
Nikiforov's spectral consecutive cycle problem and the connected-matching method
研究尼基福罗夫关于连续长度圈的开放问题，通过结合度形式的塞梅雷迪正则引理等多种方法，确定了精确常数，改进了此前结果，给出图\(G\)在一定条件下包含特定长度圈的结论。
The game chromatic number of generalized Mycielski graphs of paths and cycles
针对路径和圈的广义Mycielski图，确定其游戏色数介于4至5之间，得出M_2(P_5)与M_2(P_6)的游戏色数精确值为4，证明其不随层数和基础图阶数增长而超过5。
Spectral and Additive Combinatorial Methods for Cycles and Absorbing Sets in Lifted-Product Quantum LDPC Codes
研究提升积量子LDPC码中循环和吸收集相关问题，利用量子特定谱框架，通过对$H_X H_X^T$进行DFT块对角化，推导出广义双循环码的闭式4循环计数等结果，为该领域研究提供新方法和结论。
Enumerating Cayley digraphs on dihedral groups
该研究聚焦二面体群上Cayley有向图的CI-同构计数，借助Cauchy-Frobenius引理与自同构性质，推导了具有DCI性质的二面体群（特别是$\text{D}_{6p}$，p为大于3的素数）上非同构Cayley有向图数的显式公式。
Improved Upper Bounds on the Pebbling Numbers of the Blanuša Snarks
本文基于Hurlbert的权重函数引理，通过线性规划优化，将Blanuša蛇B₁和B₂的堆叠数上界分别改进至28和29，并重新验证了下界。
Homotopy groups and quantitative Sperner-type lemma
本文证明Sperner引理的定量形式，用边界同伦类控制完全标记单纯形数量，给出最小数量下界并计算其值，在等维和相邻维情形分别得到绝对度和九的尖锐界。
Asymptotically optimal Ramsey goodness of sparse graphs versus odd cycles and paths
本文研究稀疏图与奇数环和路径的渐近最优Ramsey良性，证明了对于奇数环n=Ω(k)足够，路径n=Ω(k^2)或附加条件n=Ω(k)。
Effective bounds for induced size-Ramsey numbers of cycles
本文通过新的宿主图构造，将环的诱导 size-Ramsey 数上界从塔型依赖改进为多项式（偶数环）和近最优指数（奇数环），并证明普通 size-Ramsey 数达到最优指数界。
Hamilton-connected cores and five cycle--wheel Ramsey numbers
通过Hamilton连通核引理和结构方法，证明了若干圈-轮Ramsey数，包括R(C_14,W_11)=27等，并给出临界染色的完整分类，所有证明均为组合证明。
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