Daniel J. Harvey, David R. Wood
We generalise a result of Corrádi and Hajnal and show that every graph with average degree at least $\tfrac{4}{3}kr$ contains $k$ vertex disjoint cycles, each of order at least $r$, as long as $k \geq 6$. This bound is sharp when $r=3$.
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We prove that for any given $\varepsilon>0$ and $d\in [0,1]$, every sufficiently large $(\varepsilon, d)$-dense graph $G$ contains for each odd integer $r$ at least $(d^r-\varepsilon)|V(G)|^r$ cycles of length $r$. Here, $G$ being $(\varepsilon, d)$-dense means that every set $X$ containing at least~$\varepsilon\,|V(G)|$ vertices spans at least $\tfrac d2\, |X|^2$ edges, and what we really count is the number of homomorphisms from an $r$-cycle into $G$. The result adresses a question of Y. Kohayakawa, B. Nagle, V. Rödl, and M. Schacht.
Balogh, Csaba, Jing and Pluhár recently determined the minimum degree threshold that ensures a $2$-coloured graph $G$ contains a Hamilton cycle of significant colour bias (i.e., a Hamilton cycle that contains significantly more than half of its edges in one colour). In this short note we extend this result, determining the corresponding threshold for $r$-colourings.
We show that every $k$-uniform hypergraph on $n$ vertices whose minimum $(k-2)$-degree is at least $(5/9+o(1))n^2/2$ contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sahueza-Matamala.
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…
We will show that for $α>0$ there is $n_0$ such that if $G$ is a graph on $n\geq n_0$ vertices such that $αn< δ(G)< (n-1)/2$, then for every $n_1+n_2+\cdots +n_l= δ(G)$, $G$ contains a disjoint union of $C_{2n_1},C_{2n_2}, \dots, C_{2n_l}$ unless $G$ has a very specific structure.
Tight Hamiltonian Cycles in Uniformly Dense $3$-Graphs
该研究确定了一致稠密3-均匀超图中紧哈密顿圈的最小代码度条件，证明1/3为渐近最优常数，同时否定回答了相关问题并明确顶点度阈值未因一致稠密性降低。
Let $G$ be a graph obtained as the union of some $n$-vertex graph $H_n$ with minimum degree $δ(H_n)\geqαn$ and a $d$-dimensional random geometric graph $G^d(n,r)$. We investigate under which conditions for $r$ the graph $G$ will a.a.s. contain the $k$-th power of a Hamilton cycle, for any choice of $H_n$. We provide asymptotically optimal conditions for $r$ for all values of $α$, $d$ and $k$. This has applications in the containment of other spanning structures, such as $F$-factors.
Hamilton Cycles in Dense Regular Digraphs and Oriented Graphs
本文证明了度数至少为 $(1/4 + \varepsilon)n$ 的稠密正则定向图包含哈密顿圈，确立了 Jackson 1981 年猜想的近似版本，并拓展了正则有向图哈密顿性的相关结果。
Near rainbow Hamilton cycles in dense graphs
针对Andersen提出的正常边着色图中近彩虹哈密顿圈问题，本文证明了满足全局n/8有界且最小度≥(1/2+ε)n的一般图存在含n-o(n)种颜色的哈密顿圈，并指出常数1/8最优。
Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the number of odd degree vertices in $G$. Then a.a.s. the following hold: (i) $G$ can be decomposed into $\lfloorΔ(G)/2\rfloor$ cycles and a matching of size $odd(G)/2$. (ii) $G$ can be decomposed into $\max\{odd(G)/2,\lceilΔ(G)/2\rceil\}$ paths. (iii) $G$ can be decomposed into $\lceilΔ(G)/2\rceil$ linear forests. Each of these bounds is best possible. We actually…
We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results. Firstly, we prove a hypergraph extension of the Erdős-Gallai Theorem: for every $δ>0$ every sufficiently lar…
In this note, we show that for any $m \in \{1,2, \dots , q +1 \}$, if $G$ is a polarity graph of a projective plane of order $q$ that has an oval, then $G$ contains a subgraph on $m + \binom{m}{2}$ vertices with $m^2+\frac{m^4}{8q} - O ( \frac{m^4}{q^{3/2} } +m )$ edges. As an application, we give the best known lower bounds on the Turán number $\mathrm{ex}(n, C_4)$ for certain values of $n$. In particular, we disprove a conjecture of Abreu, Balbuena, and Labbate concerning $\mathrm{ex}(q^2-q-2, C_4)$ where $q$ is a power of $2$.
A famous conjecture of Lovász states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.
We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy considered it for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. We show that if an $n$-vertex $3$-uniform hypergraph $H=(V,E)$ has the property that for any set of vertices $X$ and for any collection $P$ of pairs of vertices…
We consider the irreducibility of switch-based Markov chains for the approximate uniform sampling of Hamiltonian cycles in a given undirected dense graph on $n$ vertices. As our main result, we show that every pair of Hamiltonian cycles in a graph with minimum degree at least $n/2+7$ can be transformed into each other by switch operations of size at most $10$, implying that the switch Markov chain using switches of size at most $10$ is irreducible. As a proof of concept, we also show that this Markov chain is rapidly mixing on dense monotone graphs.
The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture
本文证明了Hilton猜想：对于阶数至少440的哈密顿图，若边数超过n^2/4+1，则对每个长度ℓ（3≤ℓ≤n），图中至少有n-ℓ+2个ℓ-圈。
Let G be a digraph (without parallel edges) such that every directed cycle has length at least four; let $β(G)$ denote the size of the smallest subset X in E(G) such that $G\X$ has no directed cycles, and let $γ(G)$ be the number of unordered pairs {u,v} of vertices such that u,v are nonadjacent in G. It is easy to see that if $γ(G) = 0$ then $β(G) = 0$; what can we say about $β(G)$ if $γ(G)$ is bounded? We prove that in general $β(G)$ is at most $γ(G)$. We conjecture that in fact $β(G)$ is at most $γ(G)/2$ (this would be best possible if true), and prove this conjecture in two special cases: …
Cycle Partitions in Dense Regular Digraphs and Oriented Graphs
本文证明了Jackson在1981年提出的关于大$d$-正则定向图哈密顿性的猜想，并推广证明了满足$d\geqαn$的稠密$d$-正则有向图和定向图可分别被至多$n/(d+1)$和$n/(2d+1)$个顶点不相交的圈覆盖。
Lollipops, dense cycles and chords
本文推广了Gupta-Kahn-Robertson定理，证明高最小度图中存在可收缩边得到高最小度的循环子式，并进一步证明最小度$O(k^2)$保证循环$K_k$-子式。
On sufficient conditions for spanning structures in dense graphs
该研究将稠密图中哈密顿图的鲁棒结构条件推广到环幂与次线性带宽图，解决了相关嵌入问题并恢复建立了多场景下的带宽定理。
We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil \frac{n}{2}\right\rceil$ paths, while a conjecture of Hajós states that any Eulerian graph on $n$ vertices can be decomposed into at most $\left\lfloor \frac{n-1}{2}\right\rfloor$ cycles. The Erdős-Gallai conjecture states that any graph on $n$ vertices can be decomposed into $O(n)$ cycles and edges. We show that if $G$ is a sufficiently large graph on $n$ vertices with linear mi…
Given an $n$-vertex graph $G$ with minimum degree at least $d n$ for some fixed $d > 0$, the distribution $G \cup \mathbb{G}(n,p)$ over the supergraphs of $G$ is referred to as a (random) {\sl perturbation} of $G$. We consider the distribution of edge-coloured graphs arising from assigning each edge of the random perturbation $G \cup \mathbb{G}(n,p)$ a colour, chosen independently and uniformly at random from a set of colours of size $r := r(n)$. We prove that such edge-coloured graph distributions a.a.s. admit rainbow Hamilton cycles whenever the edge-density of the random perturbation satisf…
Dense circuit graphs and the planar Turán number of a cycle
本文证明稠密电路图含大近三角剖分，由此得到环的平面Turán数上界，并验证了关于$k\ge 11$时的一个猜想。
We prove that a random cubic graph almost surely is not homomorphic to a cycle of size 7. This implies that there exist cubic graphs of arbitrarily high girth with no homomorphisms to the cycle of size 7.
Motivated by the concept of transition system investigated by Kotzig in 1968, Krivelevich, Lee and Sudakov proposed a more general notion of incompatibility system to formulate the robustness of Hamiltonicity of Dirac graphs. Given a graph $G=(V,E)$, an {\em incompatibility system} $\mathcal{F}$ over $G$ is a family $\mathcal{F}=\{F_v\}_{v\in V}$ such that for every $v\in V$, $F_v$ is a family of edge pairs in $\{\{e,e'\}: e\ne e'\in E, e\cap e'=\{v\}\}$. An incompatibility system $\mathcal{F}$ is \emph{$Δ$-bounded} if for every vertex $v$ and every edge $e$ incident with $v$, there are at mos…
Berge Hamilton cycles in a random sparsification of dense hypergraphs
研究随机过程生成的超图中Berge哈密顿环的出现时间与最小度数为2的时间一致性的条件，扩展了Johansson的结果。
Every graph of size $q$ (the number of edges) and minimum degree $δ$ is hamiltonian if $q\leδ^2+δ-1$. The result is sharp.
Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity.
We consider problems about packing and counting Hamilton $\ell$-cycles in hypergraphs of large minimum degree. Given a hypergraph $\mathcal H$, for a $d$-subset $A\subseteq V(\mathcal H)$, we denote by $d_{\mathcal H}(A)$ the number of distinct \emph{edges} $f\in E(\mathcal H)$ for which $A\subseteq f$, and set $δ_d(\mathcal H)$ to be the minimum $d_{\mathcal H}(A)$ over all $A\subseteq V(\mathcal H)$ of size $d$. We show that if a $k$-uniform hypergraph on $n$ vertices $\mathcal H$ satisfies $δ_{k-1}(\mathcal H)\geq αn$ for some $α>1/2$, then for every $\ell<k/2$ $\mathcal H$ contains $(1-o(1…
Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. There is a positive constant $\varepsilon > 0$ such that $δ(G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. There are some constants $α, β>0$ such that every two disjoint subsets $U,W$ of …
For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main purpose of this paper is to prove the following: for every s > 0 there exists n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generate…
The Kohayakawa-Nagle-Rödl-Schacht conjecture roughly states that every sufficiently large locally $d$-dense graph $G$ on $n$ vertices must contain at least $(1-o(1))d^{|E(H)|}n^{|V(H)|}$ copies of a fixed graph $H$. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. Firstly, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Secondly, we prove that a class of graphs obtained by gluing complete mu…
Spectral conditions for the existence of chorded cycles in graphs with fixed size
针对 Gould 提出的谱条件问题，本文利用谱半径刻画固定大小图包含弦圈及特定 (2k−3)-弦 (2k+1)-圈的极值条件。
Recently it was shown (by the author) that every graph of size $q$ (the number of edges) and minimum degree $δ$ is hamiltonian if $q\leδ^2+δ-1$ (arXiv:1107.2201v1). In this paper we present the exact analog of this result for dominating cycles: if $G$ is a 2-connected graph with $q\le8$ if $δ=2$ and $q\le (3(δ-1)(δ+2)-1)/2$ if $δ\ge3$, then each longest cycle in $G$ is a dominating cycle. The result is sharp in all respects.
Resilience for Loose Hamilton Cycles
本文研究随机 k-一致超图子图中的松散 Hamilton 圈，证明在给定概率阈值下其最小 d-度阈值与稠密情形一致，并说明该概率范围在部分 d 下近似紧。
On size Ramsey numbers for a pair of cycles
本文证明了一对环的尺寸拉姆齐数存在线性上界，并研究了受限尺寸拉姆齐数。
A generalization of the Hamiltonian cycle in dense digraphs
本文推广了 Hamiltonian 圈的概念，证明了在最小入度和出度至少为 $(2/3+γ)n$ 的稠密有向图中必然包含 Hamiltonian 圈的反向平方，扩展了已有结果。
Given a complete graph with positive weights on its edges, we define the weight of a subset of edges as the product of weights of the edges in the subset and consider sums (partition functions) of weights over subsets of various kinds: cycle covers, closed walks, spanning trees. We show that if the weights of the edges of the graph are within a constant factor, fixed in advance, of each other then the bulk of the partition function is concentrated on the subsets of a particularly simple structure: cycle covers with few cycles, walks that visit every vertex only few times, and spanning trees wi…
Sharp Diagonal Thresholds for Tight Hamilton Cycles in Uniformly Dense $3$-Graphs
本文在最小顶点度和最小共度数条件下，确定均匀稠密3 - 图中紧密哈密顿圈的尖锐对角阈值，证明\(d > 1/3\)且\(\alpha > f(d)\)时相关密度和度条件迫使存在紧密哈密顿圈，还给出最小共度数条件下阈值，反驳了一些猜想。
The Directed Uniform Hamilton-Waterloo Problem Involving Even Cycle Sizes
本文研究完全对称有向图的均匀分解问题，推广无向Hamilton-Waterloo问题，证明除少数例外，可分解为两个非同构均匀因子，涉及有向偶圈。
The Directed Hamilton-Waterloo Problem asks for a directed $2$-factorization of the complete symmetric digraph $K_v^*$ where there are two non-isomorphic $2$-factors. In the uniform version of the problem, factors consist of either directed $m$-cycles or $n$-cycles. In this paper, necessary conditions for a solution of this problem are given, and the problem is solved for the factors with $(m, n)\in \{(4,6),(4,8),(4,12),(4,16),(6,12)$,$(8,16),(3,5),(3,15),(5,15)\}$.
In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our r…
Goodman-type lower bounds for odd cycles I: The dense regime
该研究将古德曼的三角形下界推广到稠密 regime 中的所有奇环，通过谱展开与 Dirichlet 平均等方法证明了相关不等式，并在 Lean 4 中形式化了奇环下界定理。
Paul Erdős and László Lovász established by means of an example that there exists a maximal intersecting family of $k-$sets with approximately $(e-1)k!$ blocks. László Lovász conjectured that their example is best known example which has the maximum number of blocks. Later it was disproved. But the quest for such examples remain valid till this date. In this short note, by computing transversal size of a certain series of uniform intersecting families constructed over the cycle graph, we provide an example which has more than $(\frac{k}{2})^{k-1}$ (approximately) blocks.
In this paper we study the randomly edge colored graph that is obtained by adding randomly colored random edges to an arbitrary randomly edge colored dense graph. In particular we ask how many colors and how many random edges are needed so that the resultant graph contains a fixed number of edge disjoint rainbow Hamilton cycles. We also ask when in the resultant graph every pair of vertices is connected by a rainbow path.
Size conditions for admissible or consecutive even cycles in graphs
本文研究了图中可接受或连续偶数环的存在条件，证明了当图中无k个连续偶数长度环时，边数的上限，并在特定范围内验证了Verstraëte的猜想。
For graphs $H_1,H_2$ by $r^*(H_1,H_2)$ we denote the minimum number of edges in a graph $G$ on $r(H_1,H_2)$ vertices such that $G\to (H_1,H_2)$. We show that for each pair of natural numbers $k,n$, $k\le n$, where $k$ is odd and $n$ is large enough, we have $$r^*(C_n,C_k)=\lceil (n+1)(2n-1)/2\rceil \,.$$
The notion of robust expansion has played a central role in the solution of several conjectures involving the packing of Hamilton cycles in graphs and directed graphs. These and other results usually rely on the fact that every robustly expanding (di)graph with suitably large minimum degree contains a Hamilton cycle. Previous proofs of this require Szemerédi's Regularity Lemma and so this fact can only be applied to dense, sufficiently large robust expanders. We give a proof that does not use the Regularity Lemma and, indeed, we can apply our result to suitable sparse robustly expanding digrap…
In this paper we modify slightly Razborov's flag algebra machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges of a 4-cycle-free subgraph of the n-dimensional hypercube is at most 0.6068 times the number of its edges. We also improve the upper bound on the number of edges for 6-cycle-free subgraphs of the n-dimensional hypercube from the square root of 2 - 1 to 0.3755 times the number of its edges. Additionally, we show that if the n-dimensional hypercube is considered as a poset, then the maximum vertex density of three middle layers …
Let $\overrightarrow{P_k}$ and $\overrightarrow{C_k}$ denote the directed path and the directed cycle of order $k$, respectively. In this paper, we determine the precise maximum size of $\overrightarrow{P_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size for large $n$. For all $n$, we also determine the precise maximum size of $\overrightarrow{C_k}$-free digraphs of order $n$ as well as the extremal digraphs attaining the maximum size. In addition, Huang and Lyu [\textit{Discrete Math. 343(5) 2020}] characterized the extremal digraphs avoiding an orient…
The Multicolour Size--Ramsey Number of an Even Cycle
该研究确定了偶环的k色大小拉姆齐数，给出了其渐近阶表达式，下界源于路径相关结果，上界改进了前人的$O(k^{34}n)$估计。
A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups
本文提出一个关于带多个阿贝尔群标记的图中循环的统一半整数埃罗什-波萨定理，证明了在特定条件下循环包容量与击中所有循环的最小顶点集之间的对偶性。
Ramsey number of a cycle versus a graph of a given size
本文证明了cycle与给定大小的图的Ramsey数上限，解决了Erdős等人提出的问题。
In a graph $G$, let $μ_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $λK_{v,u}$ be the graph $(V\cup U,E)$ with $|V|=v$, $|U|=u$, and \[ μ_G(xy)=\begin{cases} λ&\mbox{if $x\in U$ and $y\in V$ or if $x\in V$ and $y\in U$}\\ 0 &\mbox{otherwise.} \\ \end{cases} \] Let $M$ be a sequence of non-negative integers $m_1,m_2,\ldots,m_n$. An $(M)$-cycle decomposition of a graph $G$ is a partition of the edge set into cycles of lengths $m_1,m_2,\ldots,m_n$. In this paper, we establish necessary and sufficient conditions for the existence of an $(M)$-cycle decomposition of $λK_{v,u}$.
The Hamilton-Waterloo problem asks for a decomposition of the complete graph into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such that $r+s=\left\lfloor\frac{v-1}{2}\right\rfloor$. If $F_{1}$ consists of $m$-cycles and $F_{2}$ consists of $n$ cycles, then we call such a decomposition a $(m,n)-$HWP$(v;r,s)$. The goal is to find a decomposition for every possible pair $(r,s)$. In this paper, we show that for odd $x$ and $y$, there is a $(2^kx,y)-$HWP$(vm;r,s)$ if $\gcd(x,y)\geq 3$, $m\geq 3$, and both $x$ and $y$ divide $v$, except possibly when $1\in\{r,s\}$.
In this work, the classical Nelson -- Hadwiger problem is studied which lies on the edge of combinatorial geometry and graph theory. It concerns colorings of distance graphs in $ {\mathbb R}^n $, i.e., graphs such that their vertices are vectors and their edges are pairs of vectors at a distance from a given set of postive numbers apart. A series of new lower bounds are obtained for the chromatic numbers of such graphs with different restrictions on the clique numbers and the girths.
Let $F$, $G$ and $H$ be simple graphs. We say $F \rightarrow (G, H)$ if for every $2$-coloring of the edges of $F$ there exists a monochromatic $G$ or $H$ in $F$. The Ramsey number $r(G, H)$ is defined as $r(G, H) = min\{|V (F)|: F \rightarrow (G, H)\}$, while the restricted size Ramsey number $r^{*}(G, H)$ is defined as $r^{*}(G, H) = min\{|E (F)|: F \rightarrow (G, H) , |V (F) | = r(G, H)\}$. In this paper we determine previously unknown restricted size Ramsey numbers $r^{*}(P_3, C_n)$ for $7 \leq n \leq 12$. We also give new upper bound $r^{*}(P_3, C_n) \leq 2n-2$ for even $n \geq 8$.
We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a complete rank, or, equivalently, of size $1 + {{n-1} \choose d}$. The discussion is restricted to the fields $F_2$ and $Q$. For $d=2$, we characterize the $n$'s for which Hamiltonian $2$-cycles exist. For $d=3$ it is shown that Hamiltonian $3$-cycles exist for infinitely many $n$'s. In general, it is shown that there always exist simple $d$-cycles of size ${{n-1} …
For given graphs $G_1, G_2,\ldots, G_n$ and any integer $j$, the size of the multipartite Ramsey number $m_j(G_1, G_2,\ldots, G_n)$ is the smallest positive integer $t$ such that any $n$-coloring of the edges of $K_{j\times t}$ contains a monochromatic copy of $G_i$ in color $i$ for some $i$, $1 \leq i \leq n$, where $K_{j\times t}$ denotes the complete multipartite graph having $j$ classes with $t$ vertices per each class. In this paper we compute the size of the multipartite Ramsey number $m_j(K_{1,2}, P_4, nK_2)$ for any $j,n\geq 2$ and $m_j(nK_2,C_7)$, for any $j\leq4$ and $n\geq 2$.
Tight Hamiltonicity from dense links of triples
该研究针对k-均匀超图的紧哈密顿环存在性问题，证明当任意k-3个顶点所在边数满足一定密度条件时超图必含紧哈密顿环，且该条件渐近最优。
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