Long-Tu Yuan
Given a graph $H$ and an integer $p$, the {\it edge blow-up} of $H$, denoted as $H^{p+1}$, is the graph obtained from replacing each edge in $H$ by a clique of size $p+1$ where the new vertices of the cliques are all different. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. In this paper, we determine the Turán numbers for edge blow-up of general graphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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…
Given a graph $H$ and an integer $p$ ($p\geq 2$), the edge blow-up $H^{p+1}$ of $H$ is the graph obtained from replacing each edge in $H$ by a clique of order $(p+1)$, where the new vertices of the cliques are all distinct. The Turán numbers for edge blow-up of matchings were first studied by Erdős and Moon. Very recently some substantial progress of the extremal graphs for $H^{p+1}$ of larger $p$ has been made by Yuan. The range of Turán numbers for edge blow-up of all bipartite graphs when $p\geq 3$ and the exact Turán numbers for edge blow-up of all non-bipartite graphs when $p\geq χ(H) +1$…
Spectral extremal graphs for edge blow-up of star forests
本文证明了当顶点数足够大时，在不含星林边膨胀副本的图中，具有最大谱半径的图即为星林边膨胀的极图。
Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponentia…
Exact extremal constructions for the inducibility of blowup graphs
对于任意图H，证明存在常数h_*(H)，使得当h≥h_*(H)且n足够大时，所有最大化H^{(h)}诱导拷贝数的n顶点图都是H的爆炸图，解决了Bollobás等人1995年提出的问题。
Spectral extremal results on edge blow-up of graphs
该研究确定了所有非二部图边膨胀的精确谱极值，给出了二部图边膨胀的渐近谱极值，推广了相关已有结果并得到了几类特殊图边膨胀的谱极值精确值。
The edge blow-up of a graph is the graph obtained from replacing each edge of it by a clique of the same size where the new vertices of the cliques are all different. Wang, Hou, Liu and Ma determined the Turán number of the edge blow-up of trees except one particular case. Answering an problem posed by them, we determined the Turán number of this particular case.
On edge-ordered graphs with linear extremal functions
该研究刻画了具有线性极值函数的连通边有序图，证明其余连通边有序图极值函数为Ω(n log n)，还拓展了边有序路径极值函数的研究范围。
Let $G$ be an $r$-uniform hypergraph of order $t$ and $ρ(G)$ is the spectral radius of $\mathcal{A}(G)$, where $\mathcal{A}(G)$ is the adjacency tensor of $G$. A blow-up of $G$ respected to a positive integer vector $(n_{1}, n_{2},\ldots,n_{t})$, denoted by $G \circ (n_{1}, n_{2},\ldots,n_{t})$, is an $r$-uniform hypergraph obtained from $G$ by replacing each vertex $j$ of $G$ with a class of vertices $V_{j}$ of size $n_{j}\ge 1$ and if $\{j_{1},j_{2},\ldots,j_{r}\}\in E(G)$, then $\{v_{i_1},v_{i_2},\ldots,v_{i_r}\}\in E(H)$ for every $v_{i_{1}}\in V_{j_{1}}, v_{i_{2}}\in V_{j_{2}},\ldots, v_{…
On the Turán number of blow-ups of $\mathcal{F}_5$
本文研究了$\mathcal{F}_5$的吹胀图的Turán数，确定了特定吹胀后的超图的精确Turán数，并展示了其具有指数级的极值构造，同时给出了所有$\mathcal{F}_5$吹胀图的Turán数的上下界。
A subgraph of an edge-coloured graph is called rainbow if all its edges have different colours. We prove a rainbow version of the blow-up lemma of Komlós, Sárközy and Szemerédi that applies to almost optimally bounded colourings. A corollary of this is that there exists a rainbow copy of any bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming that the edge-colouring of $G$ fulfills a boundedness condition that is asymptotically best possible. This has many applications beyond rainbow colourings, for example to graph decompositions, orthogonal double covers and graph …
Let a 2 to 1 directed hypergraph be a 3-uniform hypergraph where every edge has two tail vertices and one head vertex. For any such directed hypergraph F let the nth extremal number of F be the maximum number of edges that any directed hypergraph on n vertices can have without containing a copy of F. There are actually two versions of this problem: the standard version where every triple of vertices is allowed to have up to all three possible directed edges and the oriented version where each triple can have at most one directed edge. In this paper, we determine the standard extremal numbers a…
Let a 2 to 1 directed hypergraph be a 3-uniform hypergraph where every edge has two tail vertices and one head vertex. For any such directed hypergraph F let the nth extremal number of F be the maximum number of edges that any directed hypergraph on n vertices can have without containing a copy of F. There are actually two versions the directed hypergraph model for this problem: the standard version where every triple of vertices is allowed to have up to all three possible directed edges and the oriented version where each triple can have at most one directed edge. In this paper, we determine …
The typical structure of oriented graphs and digraphs with forbidden blow-up of transitive tournaments
研究禁止转置竞赛 tournaments 的吹胀体的有向图和有向图的典型结构，证明几乎所有此类图是 r 分支的，扩展了 Kuhn 等人的结果，验证了 Cherlin 的猜想。
Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^p$ of $H$ is the graph obtained from replacing each edge in $H$ by a copy of $K_p$ where the new vertices of the cliques are all distinct. Let $C_k$ and $P_k$ denote the cycle and path of length $k$, respectively. In this paper, we find sharp upper bounds for $ex(n,K_3,C_3^3)$ and the exact value for $ ex(n,K_3,P_3^3)$ and determine the graphs attaining these bounds.
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…
The edge blow-up of a graph $F$ is the graph obtained from replacing each edge in $F$ by a clique of the same size where the new vertices of the cliques are all different. In this article, we concern about the Turán problem for the edge blow-up of trees. Erdős et al. (1995) and Chen et al. (2003) solved the problem for stars. The problem for paths was resolved by Glebov (2011). Liu (2013) extended the above results to cycles and a special family of trees with the minimum degree at most two in the smaller color class (paths and proper subdivisions of stars were included in the family). In this …
A characterization of edge-ordered graphs with almost linear extremal functions
本文解决了关于边有序图 Turán 型极值问题的猜想，证明了阶色数为 2 的边有序森林具有更强的上界 $n2^{O(\sqrt{\log n})}$，并进一步猜想 $n\log^{O(1)}n$ 的上界也成立。
On extremal (almost) edge-girth-regular graphs
该研究聚焦边围长正则图（egr-图）的极值问题，证明有限平面关联图衍生的部分图为三元组$(k,g,λ)$的极值egr-图，并给出egr-图阶数的新下界。
The problem of determining extremal hypergraphs containing at most r isomorphic copies of some element of a given hypergraph family was first studied by Boros et al. in 2001. There are not many hypergraph families for which exact results are known concerning the size of the corresponding extremal hypergraphs, except for those equivalent to the classical Turan numbers. In this paper, we determine the size of extremal k-uniform hypergraphs containing at most one pair of 2-intersecting edges for k in {3,4}. We give a complete solution when k=3 and an almost complete solution (with eleven exceptio…
In this paper we deal with a Turán-type problem: given a positive integer n and a forbidden graph H, how many edges can there be in a graph on n vertices without a subgraph H? How does a graph look like if it has this extremal edge number? The forbidden graph in this article is a clique-path: a path of length k where each edge is extended to an r-clique, r >2. We determine both the extremal number and the extremal graphs for sufficiently large n.
Counting perfect edge dominating sets: extremal results and linear-time algorithms
研究树、森林和弦图类中关于完美边支配集数量的极值图，推导广义串并联图和弦图中计数完美边支配集及支配诱导匹配的线性时间算法。
The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If $G=(V_G,E_G)$ is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of $G$ is defined as $ξ^{ee}(G)=\sum_{uv\in E_G}(1/\varepsilon_G(u)+1/\varepsilon_G(v))$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$. In this paper we first introduced four edge-grafting transformations to study the mathematical p…
The cyclic edge-connectivity of a graph $G$ is the least $k$ such that there exists a set of $k$ edges whose removal disconnects $G$ into components where every component contains a cycle. We show that for graphs of minimum degree at least 3 and girth $g$ at least 4, the cyclic edge-connectivity is bounded above by $(Δ-2)g$ where $Δ$ is the maximum degree. We then prove that if the second eigenvalue of the adjacency matrix of a $d$-regular graph of girth $g\geq4$ is sufficiently small, then the cyclic edge-connectivity is $(d-2)g$, providing a spectral condition for when this upper bound on cy…
Extremal Mostar Index of Graphs with Given Number of Cut Edges
本文研究了给定割边数的连通图的Mostar指数最大值，证明最大值为k(n-2)+(n-k-1)k，并确定了极值图结构，还给出了下界和最小值的极值图特征。
Let $G$ be a connected graph. The edge revised Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $m_{0}(e|G)$ is the number of edges equidistant from both ends of $e$. In this paper, we give the minimal and the second minimal edge revised Szeged index of cacti with order $n$ and $k$ cycles, and all the graphs that achieve the minimal …
The Turán number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$ of $\mathcal{F}$ is the graph obtained by connecting the vertex $v$ to all vertices of $F$ for each $F\in \mathcal{F}$. For two integers $k\ge1$ and $r\ge2$, let $H_i$ be a graph containing a critical edge with chromatic number $r$ for any $i\in\{1,\ldots,k\}$, and let $H=\{H_1,\ldots,H_k\}+v$. In this paper, we determine $\text{ex}(n, H)$ and characterize all th…
We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.
Extremal number of edges in graphs without homeomorphically irreducible spanning trees
本文确定了不含同胚不可约生成树的k-连通图的极值边数，对于k=1和k=2给出了精确值及唯一极图。
An animal is a planar shape formed by attaching congruent regular polygons, known as tiles, along their edges. In this paper, we study extremal animals defined on regular tessellations of the plane. In 1976, Harary and Harborth studied animals in the Euclidean cases, finding extremal values for their vertices, edges, and tiles, when any one of these parameters is fixed. Here, we generalize their results to hyperbolic animals. For each hyperbolic tessellation, we exhibit a sequence of spiral animals and prove that they attain the minimal numbers of edges and vertices within the class of animals…
We prove a rainbow version of the blow-up lemma of Komlós, Sárközy and Szemerédi for $μn$-bounded edge colourings. This enables the systematic study of rainbow embeddings of bounded degree spanning subgraphs. As one application, we show how our blow-up lemma can be used to transfer the bandwidth theorem of Böttcher, Schacht and Taraz to the rainbow setting. It can also be employed as a tool beyond the setting of $μn$-bounded edge colourings. Kim, Kühn, Kupavskii and Osthus exploit this to prove several rainbow decomposition results. Our proof methods include the strategy of an alternative proo…
Extremal edge general position sets in some graphs
本文研究图的边一般位置数，刻画了边一般位置数为边数减一或等于三的图，并给出块图的尖锐上下界及若干特定块图的精确值。
The Turán type extremal problem asks to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, the spectral Turán type extremal problem asks to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without containing a linear forest as a subgraph and characterize all corresponding extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without containing $k\cdot P_3$ as a subgraph are obtained and all extremal graphs a…
A graph is minimally $k$-connected ($k$-edge-connected) if it is $k$-connected ($k$-edge-connected) and deleting arbitrary chosen edge always leaves a graph which is not $k$-connected ($k$-edge-connected). A classic result of minimally $k$-connected graph is given by Mader who determined the extremal size of a minimally $k$-connected graph of high order in 1937. Naturally, for a fixed size of a minimally $k$-(edge)-connected graphs, what is the extremal spectral radius? In this paper, we determine the maximum spectral radius for the minimally $2$-connected ($2$-edge-connected) graphs of given …
Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $λ(G)$ that linearly depends on the statistics of order-$κ$ subgraphs in a graph $G$, we consider the extremal problem of determining $λ(n,\mathcal{G})$, the maximum of $λ(G)$ over all admissible graphs $G$ of order $n$. We call the problem perfectly $B$-stable for a graph $B$ if there is a constant $C$ such that every admissible graph $G$ of order $n\ge C$ can be made into a blow-up of $B$ by changing at most $C(λ(n,\mathcal{G})-λ(G)){n\choose2}$ adjacencies. As special cases, this property describes all almost extrem…
In 1965 Erdős conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative for k=3 and n is large enough.
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…
Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.
We prove the extremal function for $K_9^=$ minors, where $K_9^=$ denotes the complete graph $K_9$ with two edges removed. In particular, we show that any graph with $n$ vertices and at least $6n - 20$ edges either contains a $K_9^=$ minor or is isomorphic to a graph obtained from disjoint copies of $K_8$ and $K_{2, 2, 2, 2, 2}$ by identifying cliques of size 5. We utilize computer assistance to prove one of our lemmas.
Let k>=3 be an integer, we prove that the maximum induced density of the k-vertex directed star in a directed graph is attained by an iterated blow-up construction. This confirms a conjecture by Falgas-Ravry and Vaughan, who proved this for k=3, 4. This question provides the first known instance of density problem for which one can prove extremality of an iterated blow-up construction. We also study the inducibility of complete bipartite digraphs and discuss other related problems.
The edge Szeged index and edge-vertex Szeged index of a graph are defined as $Sz_{e}(G)=\sum\limits_{uv\in E(G)}m_{u}(uv|G)m_{v}(uv|G)$ and $Sz_{ev}(G)=\frac{1}{2} \sum\limits_{uv \in E(G)}[n_{u}(uv|G)m_{v}(uv|G)+n_{v}(uv|G)m_{u}(uv|G)],$ respectively, where $m_{u}(uv|G)$ (resp., $m_{v}(uv|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $n_{u}(uv|G)$ (resp., $n_{v}(uv|G)$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), respectively. A …
The rainbow Turán number $\mathrm{ex}^*(n,H)$ of a graph $H$ is the maximum possible number of edges in a properly edge-coloured $n$-vertex graph with no rainbow subgraph isomorphic to $H$. We prove that for any integer $k\geq 2$, $\mathrm{ex}^*(n,C_{2k})=O(n^{1+1/k})$. This is tight and establishes a conjecture of Keevash, Mubayi, Sudakov and Verstraëte. We use the same method to prove several other conjectures in various topics. First, we prove that there exists a constant $c$ such that any properly edge-coloured $n$-vertex graph with more than $cn(\log n)^4$ edges contains a rainbow cycle. …
We study the equality of the extremal Betti numbers of the binomial edge ideal $J_G$ and those of its initial ideal ${\rm in}(J_G)$ of a closed graph $G$. We prove that in some cases there is an unique extremal Betti number for ${\rm in}(J_G)$ and as a consequence there is an unique extremal Betti number for $J_G$ and these extremal Betti numbers are equal
Spectral Extremal Graphs of Planar Graphs with Fixed Size
本文研究固定边数外平面图与平面图的谱极值问题，证明外平面图极值图为星图，并给出平面图在边数奇偶不同时的谱极值图结构。
Changes in the Seidel energy of blow-up graphs under edge deletion
研究边删除下边膨胀图的Seidel能量变化，建立独立集膨胀图框架内的结构准则，给出Turán图和完全多部图边删除时Seidel能量变化情况，为Tian等人的问题提供肯定答案。
Revisiting Extremal Graphs Having No Stable Cutsets
本文填补了Le和Pfender关于边数为2n-3且无稳定割集的极图递归构造证明中的漏洞，完善了对Caro猜想相关极图结构的论证。
Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in\mathbb{N}$, let $\mathcal{F}_k$ and $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n…
In this paper extremal problems for uniform hypergraphs are studied in the general setting of hereditary properties. It turns out that extremal problems about edges are particular cases of a general analyic problem about a recently introduced graph parameter. The paper builds a basis for the systematic study of this parameter and illustrates a range of various proof tools. It is shown that extremal problems about the number of edges of uniform hypergraphs are asymptotically equivalent to extremal problems about the largest eigenvalue; this result is new even for 2-graphs. Several concrete prob…
We investigate extremal functions ex_e(F,n) and ex_i(F,n) counting maximum numbers of edges and maximum numbers of vertex-edge incidences in simple hypergraphs H which have n vertices and do not contain a fixed hypergraph F; the containment respects linear orderings of vertices. We determine both functions exactly if F has only distinct singleton edges or if F is one of the 55 hypergraphs with at most four incidences (we give proofs only for six cases). We prove some exact formulae and recurrences for the numbers of hypergraphs, simple and all, with n incidences and derive rough logarithmic as…
Strong Edge Geodetic Problem on Complete Multipartite Graphs and some Extremal Graphs for the Problem
本文研究强边测地线数，确定完全多部图的该参数，刻画 ${\rm sg_e}(G)=n(G)$ 的图，并给出笛卡尔积 $P_n\square K_m$ 的结果，纠正文献错误。
We prove that every graph with $n$ vertices and at least $5n-8$ edges contains the Petersen graph as a minor, and this bound is best possible. Moreover we characterise all Petersen-minor-free graphs with at least $5n-11$ edges. It follows that every graph containing no Petersen minor is 9-colourable and has vertex arboricity at most 5. These results are also best possible.
Spectral extremal results for triangle-free graphs with chromatic number at least four
本文研究色数至少为四的三角图自由图的谱极值问题，确定了当n足够大时，G(2,4)是格罗茨sch图的吹胀图，并与另一研究结果一致。
This note answers extremal questions like: what is the maximum number of edges in a graph of order n, which belongs to some hereditary property. The same question is answered also for the spectral radius and other similar parameters.
Given integers $r$ and $b$ with $1 \leq b \leq r$, a finite simple connected graph $G$ for which ${\rm reg}(S/I(G)) = r$ and the number of extremal Betti numbers of $S/I(G)$ is equal to $b$ will be constructed.
The spectral radii and extremal graphs of two types of minimal graphs
本文通过耳分解和边交换技术，确定了极小匹配覆盖双图和极小因子临界图的最大谱半径并刻画了极值图。
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。
Extremal Bounds on Peripherality Measures
本文研究了网络中顶点与边的边缘性度量，改进了边边缘性、边和边缘性及 Trinajstić 指数在 n 顶点图上的渐近上界，扩展至二部图、树和固定直径图，并反驳了 Furtula 的两个猜想。
The "edge polytope" of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k =2, 3, 5 we determine the maximum number of vertices of k-neighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.
This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct extremal graphs that achieve the edge bounds in all cases. We further establish uniqueness of these extremal graphs whenever they are unique.
A counterexample to the Fang--Lin conjecture: Edge and spectral extremality diverge near a Turán graph
本文构造反例 $F=K_1\vee\mu(K_3)$，证明存在边色临界图使得边极值与谱极值不一致，即无图能同时最大化边数和谱半径。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。