Zhiyang He
In this paper, we prove $\text{ex}(n, C_{2k})\le (16\sqrt{5}\sqrt{k\log k} + o(1))\cdot n^{1+1/k}$. We improved on Bukh--Jiang's method used in their 2017 publication, thereby reducing the best known upper bound by a factor of $\sqrt{5\log k}$.
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We study sufficient conditions for Hamiltonian cycles in hypergraphs, and obtain both Turán- and Dirac-type results. While the Turán-type result gives an exact threshold for the appearance of a Hamiltonian cycle in a hypergraph depending only on the extremal number of a certain path, the Dirac-type result yields a sufficient condition relying solely on the minimum vertex degree.
We give a simple geometric interpretation of an algebraic construction of Wenger that yields $n$-vertex graphs with no cycle of length $4$, $6$ or $10$ and close to the maximum number of edges.
We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.
Let $ρ(G)$ denote the number of convex cycles of a simple graph G of order n, size m, and girth 3 <= g <=n. It is proved that $ρ(G) \leq \frac{n}{g}(m-n+1)$ and that equality holds if and only if G is an even cycle or a Moore graph. The equality also holds for a possible Moore graph of diameter 2 and degree 57 thus giving a new characterization of Moore graphs.
The \emph{blow-up} of a graph $H$ is the graph obtained from replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. Erdős et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blow-ups of paths. We determined the extremal number and found the extremal graphs for the blow-ups of cycles and a large class of trees, when $n$ is sufficiently large. This generalizes their results. The additional aim of our note is to draw attention to a powerful tool…
An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdös et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we dete…
In the 1960s, Erdős and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by Bollobás, it is very difficult for general $k$. Recently, Tait and Tobin [J. Combin. Theory Ser. B, 2017] confirmed a famous conjecture on maximum spectral radius of $n$-vertex planar graphs. Motivated by the above results, we consider two extremal problems on planar graphs without $k$ edge-disjoint cycles. We first determine the maximum number of edges in a plan…
We prove an upper bound for the number of edges a C4-free graph on q^2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q.
We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s \ge 2$ and sufficiently large $n$. Moreover, for $s\in\{2,3\}$ and any integer $n$ we obtain the maximum number of cycles of length $2s$ in an $n$-vertex $C_{2s+2}$-free bipartite graph.
We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a disproof of a conjecture of Erdős from 1970s which asserts that $f(n)=\frac12 n^{3/2}+\frac14 n+o(n)$.
We describe the C_{2k+1}-free graphs on n vertices with maximum number of edges. The extremal graphs are unique except for n = 3k-1, 3k, 4k-2, or 4k-1. The value of ex(n,C_{2k+1}) can be read out from the works of Bondy, Woodall, and Bollobas, but here we give a new streamlined proof. The complete determination of the extremal graphs is also new. We obtain that the bound for n_0(C_{2k+1}) is 4k in the classical theorem of Simonovits, from which the unique extremal graph is the bipartite Turan graph.
Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} + \frac{1}{3(2k-2)}} 2^n).$$
As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approac…
In this paper we estimate the planar Turán number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when $H=C_5$, and study the cases when $G$ is bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly new applications or variants of the "contribution method" introduced by Ghosh, Győri, Martin, Paulos and Xiao in arXiv:2004.14094.
For a family $\mathcal{F}$ of graphs, let $ex(n,\mathcal{F})$ denote the maximum number of edges in an $n$-vertex graph which contains none of the members of $\mathcal{F}$ as a subgraph. A longstanding problem in extremal graph theory asks to determine the function $ex(n,\{C_3,C_4\})$. Here we give a new construction for dense graphs of girth at least five with arbitrary number of vertices, providing the first improvement on the lower bound of $ex(n,\{C_3,C_4\})$ since 1976. As a corollary, this yields a negative answer to a problem in Chung-Graham [3].
In this paper we define cycle-star graph CSk,n-k to be a graph on n vertices consisting of the cycle of length k and n-k leafs appended to the same vertex of the cycle. Also, we define cycle-path graph CPk,n-k to be a graph on n vertices consisting of the cycle of length k and of path on n-k vertices whose one end is linked to a vertex on a cycle. We establish that cycle-star graph CS3,n-3 is the only maximal graph with respect to additively weighted Harary index among all unicyclic graphs on n vertices, while cycle-path graph CP3,n-3 is the only minimal unicyclic graph (here n must be at leas…
Denote by $tC_\ell$ the disjoint union of $t$ cycles of length $\ell$. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\ge f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erdős [Arch. Math. 13 (1962) 222--227] as well as the research on $ex(n,C_{2\ell+1})$ of Füredi an…
Given graphs $G$ and $H$, the generalized Turán number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of $G$. In this paper, we obtain an asymptotic upper bound on ${\rm ex}(CT_n,C_{2l})$ for any $n \ge 3$ and $l\geq2$, where $C_{2l}$ is the cycle of length $2l$ and $CT_n$ is the complete transposition graph which is defined as the Cayley graph on the symmetric group ${\rm S}_n$ with respect to the set of all transpositions of ${\rm S}_n$.
The Turán type numbers for graphs without 3-cycles and 4-cycles are determined for vertex numbers from 40 to 49 inclusive. Hence, now, 43 of the first 50 numbers of OEIS A006856 are known. Estimates for the remaining seven numbers are presented.
Spectral extremal problem of the $p$th power of cycles
本文确定了足够大顶点数下，不含环的 $p$ 次幂作为子图的图中，边数和谱半径达到最大值时的唯一极值图。
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…
Extremal graphs with cycle lengths in a fixed residue class
本文针对模k剩余类的2-连通圈受限图，确定了非零剩余类对应的最大边数及达到该值的图，结合已有结果解决了所有剩余类的相关极值问题。
Let C be an extremal binary doubly even self-dual code of length n and s(C) denote the largest integer t such that the support design of C holds a t-design for some weight. In this paper, we prove s(C) \leq 7.
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。
A tight cycle in an $r$-uniform hypergraph $\mathcal{H}$ is a sequence of $\ell\geq r+1$ vertices $x_1,\dots,x_{\ell}$ such that all $r$-tuples $\{x_{i},x_{i+1},\dots,x_{i+r-1}\}$ (with subscripts modulo $\ell$) are edges of $\mathcal{H}$. An old problem of V. Sós, also posed independently by J. Verstraëte, asks for the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices which has no tight cycle. Although this is a very basic question, until recently, no good upper bounds were known for this problem for $r\geq 3$. Here we prove that the answer is at most $n^{r-1+o(1)}$, which …
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 $2$.
We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $Γ$ the generalized Turán density $π_F(Γ)$ denotes the density of a maximum subgraph of $Γ$, which contains no copy of~$F$. Extending classical Turán type results for odd cycles, we show that $π_{F}(Γ)=1/2$ provided $F$ is an odd cycle and $Γ$ is a sufficiently pseudorandom graph. In particular, for $(n,d,λ)$-graphs $Γ$, i.e., $n$-vertex, $d$-regular graphs with all non-trivial eigenvalues in the interval $[-λ,λ]$, our result holds for odd cycles of length $\ell$, provided \[ λ^{\ell-2}\ll \frac{d^{\ell-1}}n…
Upper Bounds on Turán Densities via Extremal Set Theory
本文系统展示了超图 Turán 问题与极值集合论之间的联系，通过构造一致超图族，将其 Turán 密度的上界归结为极值集合论中的经典问题。
Burr and Erdős conjectured that for each $k,\ell \in \mathbb Z^+$ such that $k \mathbb Z + \ell$ contains even integers, there exists $c_k(\ell)$ such that any graph of average degree at least $c_k(\ell)$ contains a cycle of length $\ell$ mod $k$. This conjecture was proved by Bollobás, and many successive improvements of upper bounds on $c_k(\ell)$ appear in the literature. In this short note, for $1 \leq \ell \leq k$, we show that $c_k(\ell)$ is proportional to the largest average degree of a $C_{\ell}$-free graph on $k$ vertices, which determines $c_k(\ell)$ up to an absolute constant. In p…
Sharp linear Turán remainders for cycle and edge extremality
本文针对Morrison等人提出的问题，证明对每个色数，不存在同时最大化边数和圈数的H-自由图，还确定了最优常数γ_r，明确了圈极大H-自由图为边极值图的条件。
We derive a number of extremal and Ramsey stability results for cycles.
We introduce a containment relation of hypergraphs which respects linear orderings of vertices and investigate associated extremal functions. We extend, by means of a more generally applicable theorem, the n.log n upper bound on the ordered graph extremal function of F=({1,3}, {1,5}, {2,3}, {2,4}) due to Z. Furedi to the n.(log n)^2.(loglog n)^3 upper bound in the hypergraph case. We use Davenport-Schinzel sequences to derive almost linear upper bounds in terms of the inverse Ackermann function. We obtain such upper bounds for the extremal functions of forests consisting of stars whose all cen…
Let $G$ be a graph on $n$ vertices, labeled $v_1,\ldots,v_n$ and $π$ be a permutation on $[n]:=\{1,2,\cdots, n\}$. Suppose that each pebble $p_i$ is placed at vertex $v_{π(i)}$ and has destination $v_i$. During each step, a disjoint set of edges is selected and the pebbles on each edge are swapped. Let $rt(G, π)$, the routing number for $π$, be the minimum number of steps necessary for the pebbles to reach their destinations. Li, Lu, and Yang prove that $rt(C_n, π)\le n-1$ for any permutation on $n$-cycle $C_n$ and conjecture that for $n \geq 5$, if $rt(C_n, π) = n-1$, then $π= (123\cdots n)$ …
Extremal problems involving independent sets are much studied. Two of the most important extremal problems in this context are concerned with the sharp upper bounds for the number of independent sets of fixed size and the independence number. In literature, these sharp upper bounds are derived in completely different contexts. In this paper, we show that both of these sharp upper bounds can be derived by considering Lex graphs. More exactly, they depend on two parameters of a sequence defined on them.
An embedding of a graph in $3$-space is linkless if for every two disjoint cycles there exists an embedded ball that contains one of the cycles and is disjoint from the other. We prove that every bipartite linklessly embeddable (simple) graph on $n\ge5$ vertices has at most $3n-10$ edges, unless it is isomorphic to the complete bipartite graph $K_{3,n-3}$.
The Erdős--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs with extremal examples $H_{n,k,t}$ and $H_{n,k,2}$. In this note, we generalize the result of Kopylov to bound the number of $s$-cliques in a graph with circumference less than $k$. Furthermore, we show that the same extremal examples that maximize the number of edges also maximize the number of cliques of any fixed size. Finally, we obtain the extremal number of…
Extremal hypergraphs without generalized 4-cycles
本文证明了Erdős关于不含广义4-环的极值超图边数的猜想，确定了所有极值配置，并解决了Mubayi的猜想。
Extremal graphs for odd-ballooning of bipartite graphs
本文确定了当t≥5时二部图的奇气球图的Turán数的范围，并推导出星图、路径和偶圈的奇气球图的Turán数。
The optimality of the Erdős-Rado theorem for pairs is witnessed by the colouring $Δ_κ: [2^κ]^2 \rightarrow κ$ recording the least point of disagreement between two functions. This colouring has no monochromatic triangles or, more generally, odd cycles. We investigate a number of questions investigating the extent to which $Δ_κ$ is an \emph{extremal} such triangle-free or odd-cycle-free colouring. We begin by introducing the notion of $Δ$-regressive and almost $Δ$-regressive colourings and studying the structures that must appear as monochromatic subgraphs for such colourings. We also consider …
Let $G$ be a non-bipartite graph which does not contain any odd cycle of length at most $2k+1$. In this paper, we determine the maximum $Q$-index of $G$ if its order is fixed, and the corresponding extremal graph is uniquely characterized. Moreover, if the size of $G$ is given, the maximum $Q$-index of $G$ and the unique extremal graph are also proved.
Extremal Numbers of Hypergraph Suspensions of Even Cycles
本文将不含偶圈的二部图极值问题扩展至三元系统，证明了任意顶点链接中不含 $C_6$ 的三元组最大数量级为 $n^{7/3}$，并构造了边数为 $n^{4/3}$ 的无 $C_6$ 稠密二部图新族。
A game starts with the empty graph on $n$ vertices, and two player alternate adding edges to the graph. Only moves which do not create a triangle are valid. The game ends when a maximal triangle-free graph is reached. The goal of one player is to end the game as soon as possible, while the other player is trying to prolong the game. With optimal play, the length of the game (number of edges played) is called the $K_3$ game saturation number. In this paper we prove an upper bound for this number.
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。
Extremal digraphs on Meyniel-type condition for hamiltonian cycles in balanced bipartite digraphs
本文研究了平衡二分有向图中哈密顿圈的Meyniel型条件，刻画了满足度条件 $d(u)+d(v)\ge 3a$ 的极值有向图的两类结构，并证明了在条件放宽至 $3a-1$ 时图可迹，且下界紧。
A set $W\subseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $β(G)$. In this paper, it is proved that in a connected graph $G$ of order $n$ which has a cycle, $β(G)\leq n-g(G)+2$, where $g(G)$ is the length of a shortest cycle in $G$, and the equality holds if and only if $G$ is a cycle, a complete graph or a complete bipartite g…
Extremal spectral results of planar graphs without vertex-disjoint cycles
本文研究不含顶点不交圈的平面图的极值谱与边数问题，确定了1≤t≤2时相关谱极值及唯一极值图，并得到2C₄和2𝒞对应的平面图Turán数精确值，解决了Li的一个猜想。
Let $EG_r(n,k)$ denote the maximum number of edges in an $n$-vertex $r$-uniform hypergraph with no Berge cycles of length $k$ or longer. In the first part of this work, we have found exact values of $EG_r(n,k)$ and described the structure of extremal hypergraphs for the case when $k-2$ divides $n-1$ and $k\geq r+3$. In this paper we determine $EG_r(n,k)$ and describe the extremal hypergraphs for all $n$ when $k\geq r+4$.
An extremal graph for a given graph $H$ is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $s,t$ be integers and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ cycles of odd lengths at least 5 which intersect in exactly one common vertex. Erdős et al. (1995) determined the extremal graphs for $H_{s,0}$. Recently, Hou et al. (2016) determined the extremal graphs for $H_{0,t}$, where the $t$ cycles have the same odd length $q$ with $q\ge 5$. In this paper, we further determine the extremal graphs for $H_{s,t}$ with $s\ge 0$ and $t\g…
Extremal graphs for odd-ballooning of paths and stars
本文利用Simonovits逐步归纳引理，统一确定了q≥3时星与路径的奇膨胀图的极值图，完善了此前分情况的相关研究结果。
Spectral extremal hypergraphs without long Berge cycles
本文确定了不含长度至少为$k$的Berge圈的$r$-均匀超图的最大谱半径，并给出了奇偶两种情形下的唯一极值结构及渐近公式。
The subpath number of cactus graphs
本文研究仙人掌图的子路径数，刻画了具有n个顶点和k个圈的极大与极小仙人掌图结构，并证明子路径数与子树数及Wiener指数均不相关。
Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti numbers are given.
On 2-connected graphs without cycles of length 1 modulo 3
本文研究了2-连通图中无长度模3余1的环的最大边数，给出了精确极值界和极值图的完全刻画，并推广到模4余2的情形。
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…
The Randic (connectivity) index is one of the most successful molecular descriptors in structure-property and structure-activity relationships studies. J. Gao found the sharp upper bound for the Randic index of apex trees. In this paper, we proved that k-apex trees are not regular graphs for k\ge2 and proposed a sharp upper bound for the Randic index of k-apex trees for k>1.
On $2$-connected graphs avoiding cycles of length $0$ modulo $4$
本文研究不含长度模4余0的圈的2-连通图的边数极值，证明了其至多有⌊(3n-1)/2⌋条边，并构造无穷多极值例子证明界是紧的。
A stability theorem for Berge Hamiltonian cycles under a minimum degree condition
该研究针对r-均匀超图，在最小度条件下，利用Salia的Pósa型定理得出不含Berge哈密顿圈的超边数极值上界，并证明近极值实例属于两类非哈密顿构造，为Berge哈密顿圈的稳定性提供了理论支撑。
Counterexamples to an Extremal Conjecture for Random Cycle-Factors
本文推翻了关于随机循环因子中循环数期望的最大化猜想，针对d≥3和n=kd的情况构造了反例，并指出在d=2时猜想成立，形成二元区分。
We study the problem of maximizing Betti numbers of simplicial complexes. We prove an upper bound of 1.32^n for the sum of Betti numbers of any n-vertex flag complex and 1.25^n for the independence complex of a triangle-free graph. These findings imply upper bounds for the Betti numbers of various related classes of spaces, including the neighbourhood complex of a graph. We also make some related observations.
Ramsey multiplicity and extremal colorings for odd cycles
本文证实了关于奇圈阈值Ramsey重数的猜想，确定$m(C_k)=(k-1)!/2$，并刻画了$K_{2k-1}$的所有极值着色，为第二个确定$m(H)$的图族。
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