Edge distribution of graphs with few induced copies of a given graph

On edge-decomposition of cubic graphs into copies of the double-star with four edges

A tree containing exactly two non-pendant vertices is called a double-star. Let $k_1$ and $k_2$ be two positive integers. The double-star with degree sequence $(k_1+1, k_2+1, 1, \ldots, 1)$ is denoted by $S_{k_1, k_2}$. If $G$ is a cubic graph and has an $S$-decomposition, for a double-star $S$, then $S$ is isomorphic to $S_{1,1}$, $S_{1,2}$ or $S_{2,2}$. It is known that a cubic graph has an $S_{1,1}$-decomposition if and only if it contains a perfect matching. In this paper we study the $S_{1,2}$-decomposition of cubic graphs. First, we present some necessary conditions for the existence of …

Rainbow copies of $F$ in families of $H$

We study the following problem. How many distinct copies of $H$ can an $n$-vertex graph $G$ have, if $G$ does not contain a rainbow $F$, that is, a copy of $F$ where each edge is contained in a different copy of $H$? The case $H=K_r$ is equivalent to the Turán problem for Berge hypergraphs, which has attracted several researchers recently. We also explore the connection of our problem to the so-called generalized Turán problems. We obtain several exact results. In the particularly interesting symmetric case where $H=F$, we completely solve the case $F$ is the 3-edge path, and asymptitically so…

On edges not in monochromatic copies of a fixed bipartite graph

Let $H$ be a fixed graph. Denote $f(n,H)$ to be the maximum number of edges not contained in any monochromatic copy of $H$ in a 2-edge-coloring of the complete graph $K_n$, and $ex(n,H)$ to be the {\it Turán number} of $H$. An easy lower bound shows $f(n,H)\ge ex(n,H)$ for any $H$ and $n$. In \cite{KS2}, Keevash and Sudakov proved that if $H$ is an edge-color-critical graph or $C_4$, then $f(n,H)= ex(n,H)$ holds for large $n$, and they asked if this equality holds for any graph $H$ when $n$ is sufficiently large. In this paper, we provide an affirmative answer to this problem for an abundant i…

Packing a number of copies of a $(p,\,q)$-graph

Let $k,p,q$ be three positive integers. A graph $G$ with order $n$ is said to be $k$-placeable if there are $k$ edge disjoint copies of $G$ in the complete graph on $n$ vertices. A $(p,\,q)$-graph is a graph of order $p$ with $q$ edges. Packing results have proved useful in the study of the complexity of graph properties. Bollobás et al. investigated the $k$-placeable of $(n,\,n-2)$-graphs and $(n,\,n-1)$-graphs with $k=2$ and $k=3$. Motivated by their results, this paper characterizes $(n,\,n-1)$-graphs with girth at least $9$ which are $4$-placeable. We also consider the $k$-placeable of $(n…

避免彩虹$K_4$拷贝的边着色的最大数量

Maximum number of edge colorings avoiding rainbow copies of $K_4$

本文证明了对于$r\geq 12$及足够大的$n$顶点图，无彩虹$K_4$的$r$-边着色数上界为$r^{ex(n,K_4)}$，并刻画了取等条件，改进了前人结果并证实了相关猜想。

Properly colored and rainbow copies of graphs with few cherries

Let G be an n-vertex graph that contains linearly many cherries (i.e., paths on 3 vertices), and let c be a coloring of the edges of the complete graph K_n such that at each vertex every color appears only constantly many times. In 1979, Shearer conjectured that such a coloring c must contain a properly colored copy of G. We establish this conjecture in a strong form, showing that it holds even for graphs G with O(n^(4/3)) cherries and moreover this bound on the number of cherries is best possible up to a constant factor. We also prove that one can find a rainbow copy of such G in every edge-c…

包含三角形的图中单色副本的一个性质

A property on monochromatic copies of graphs containing a triangle

本文证明了每个包含$K_3$且至少有四条边的图都不是strongly common，推广了Jagger等人关于包含$K_4$的图不是common的著名定理。

在具有规定边数的图中最大化固定图的副本数

Maximizing copies of a fixed graph in graphs with a prescribed number of edges

研究在规定边数的图中最大化固定图副本数的问题，通过相关结果确定了特定图\(H\)的\(\operatorname{emb}(H,{\binom{n}{2}})\)值，还证明除匹配图外，\(K_n\)是使嵌入数最大的唯一图，刻画了\(K_n\)达最大嵌入数的图。

Homogeneous edge-disjoint $K_{2s}$ and $T_{st,t}$ unions

Let $r>2$ and $σ\in(0,r-1)$ be integers. We require $t<2s$, where $t=2^{σ+1}-1$ and $s=2^{r-σ-1}$. Generalizing a known $\{K_4,T_{6,3}\}$-ultrahomogenous graph $G_3^1$, we find that a finite, connected, undirected, arc-transitive graph $G_r^σ$ exists each of whose edges is shared by just two maximal subgraphs, namely a clique $X_0=K_{2s}$ and a $t$-partite regular-Turán graph $X_1=T_{st,t}$ on $s$ vertices per part. Each copy $Y$ of $X_i$ ($i=0,1$) in $G_r^σ$ shares each edge with just one copy of $X_{1-i}$ and all such copies of $X_{1-i}$ are pairwise distinct. Moreover, $G_r^σ$ is an edge-di…

Ramsey numbers for multiple copies of hypergraphs

In this paper, for sufficiently large $n$ we determine the Ramsey number $R(G,nH)$ where $G$ is a $k$-uniform hypergraph with the maximum independent set that intersects each of the edges in $k-1$ vertices and $H$ is a $k$-uniform hypergraph with a vertex so that the hypergraph induced by the edges containing this vertex is a star. There are several examples for such $G$ and $H$, among them are any disjoint union of $k$-uniform hypergraphs involving loose paths, loose cycles, tight paths, tight cycles with a multiple of $k$ edges, stars, Kneser hypergraphs and complete $k$-uniform $k$-partite …

A proof of Ringel's Conjecture

A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the decomposition of complete graphs into edge-disjoint copies of a tree. It says that any tree with $n$ edges packs $2n+1$ times into the complete graph $K_{2n+1}$. In this paper, we prove this conjecture for large $n$.

生成子图的彩虹拷贝

Rainbow copies of spanning subgraphs

本文研究了 $n$ 顶点边着色随机图空间 $G_{n,p}^{[κ]}$ 中生成子图 $H$ 的彩虹拷贝存在阈值，提供了概率 $p$ 和颜色数 $κ$ 的下界，以高概率证明此类拷贝的存在性。

Few $T$ copies in $H$-saturated graphs

A graph is $F$-saturated if it is $F$-free but the addition of any edge creates a copy of $F$. In this paper we study the quantity $\mathrm{sat}(n, H, F)$ which denotes the minimum number of copies of $H$ that an $F$-saturated graph on $n$ vertices may contain. This parameter is a natural saturation analogue of Alon and Shikhelman's generalized Turán problem, and letting $H = K_2$ recovers the well-studied saturation function. We provide a first investigation into this general function focusing on the cases where the host graph is either $K_s$ or $C_k$-saturated. Some representative interestin…

The Turan Number of Disjoint Copies of Paths

The Turán number of a graph $H$, $ex(n,H)$, is the maximum number of edges in a simple graph of order $n$ which does not contain $H$ as a subgraph. Let $k\cdot P_3$ denote $k$ disjoint copies of a path on $3$ vertices. In this paper, we determine the value $ex(n, k\cdot P_3)$ and characterize all extremal graphs. This extends a result of Bushaw and Kettle [N. Bushaw and N. Kettle, Turán Numbers of multiple and equibipartite forests, Combin. Probab. Comput., 20(2011) 837-853.], which solved the conjecture proposed by Gorgol in [I. Gorgol. Turán numbers for disjoint copies of graphs. {\it Graphs…

On a {K_4,K_{2,2,2}}-ultrahomogeneous graph

The existence of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ultrahomogeneous graph $G$ is established, (i.e. each isomorphism between two copies of $K_4$ or $K_{2,2,2}$ in $G$ extends to an automorphism of $G$), with the 42 ordered lines of the Fano plane taken as vertices. This graph $G$ can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of $K_4$ and as the edge-disjoint union of 21 induced copies of $K_{2,2,2}$, with no more copies of $K_4$ or $K_{2,2,2}$ existing in $G$. Moreover, each edge of $G$ is shared by exactly one copy of $K_4$ and one of $K_{2,2,2…

Asymptotically optimal $K_k$-packings of dense graphs via fractional $K_k$-decompositions

Let $H$ be a fixed graph. A {\em fractional $H$-decomposition} of a graph $G$ is an assignment of nonnegative real weights to the copies of $H$ in $G$ such that for each $e \in E(G)$, the sum of the weights of copies of $H$ containing $e$ in precisely one. An {\em $H$-packing} of a graph $G$ is a set of edge disjoint copies of $H$ in $G$. The following results are proved. For every fixed $k > 2$, every graph with $n$ vertices and minimum degree at least $n(1-1/9k^{10})+o(n)$ has a fractional $K_k$-decomposition and has a $K_k$-packing which covers all but $o(n^2)$ edges.

关于三个2-因子副本的打包唯一性

On uniqueness of packing of three copies of 2-factors

本文完全解决了2-正则图三个副本打包的唯一性问题，给出了无打包、唯一打包和至少两种打包的完整分类。

Minimum Rainbow $H$-Decompositions of Graphs

Given graphs $G$ and $H$, we consider the problem of decomposing a properly edge-colored graph $G$ into few parts consisting of rainbow copies of $H$ and single edges. We establish a close relation to the previously studied problem of minimum $H$-decompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of $H$ and single edges.

Spanning $F$-cycles in random graphs

We extend a recent argument of Kahn, Narayanan and Park (Proceedings of the AMS, to appear) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold…

Stability and Erdős--Stone type results for $F$-free graphs with a fixed number of edges

A fundamental problem of extremal graph theory is to ask, 'What is the maximum number of edges in an $F$-free graph on $n$ vertices?' Recently Alon and Shikhelman proposed a more general, subgraph counting, version of this question. They considered the question of determining the maximum number of copies of a fixed graph $T$ in an $F$-free graph on $n$ vertices. In this more general context, where we are no longer counting edges, it is also natural to ask what is the maximum number of copies of $T$ in an $F$-free graph with $m$ edges and no restriction on the number of vertices. Frohmader, in …

Efficiently finding low-sum copies of spanning forests in zero-sum complete graphs via conditional expectation

For a fixed positive $ε$, we show the existence of a constant $C_ε$ with the following property: Given a $\pm1$-edge-labeling $c:E(K_n)\to \{ -1,1\}$ of the complete graph $K_n$ with $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$ of maximum degree $Δ$, one can determine in polynomial time an isomorphic copy $F'$ of $F$ in $K_n$ with $|c(E(F'))|\leq \left(\frac{3}{4}+ε\right)Δ+C_ε.$ Our approach is based on the method of conditional expectation.

On the inducibility problem for random Cayley graphs of abelian groups with a few deleted vertices

Given a $k$-vertex graph $H$ and an integer $n$, what are the $n$-vertex graphs with the maximum number of induced copies of $H$? This question is closely related to the inducibility problem introduced by Pippenger and Golumbic in 1975, which asks for the maximum possible fraction of $k$-vertex subsets of an $n$-vertex graph that induce a copy of $H$. Huang, Lee and the first author proved that for a random $k$-vertex graph $H$, almost surely the $n$-vertex graphs maximizing the number of induced copies of $H$ are the balanced iterated blow-ups of $H$. In this paper, we consider the case where…

Generalized rainbow Turán problems

Alon and Shikhelman initiated the systematic study of the following generalized Turán problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Turán number of $F$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $F$. The study of rainbow Turán problems was initiated by Keevash, Mubayi, Sudakov and Verstraëte. Motivated by the above problems, we study the followi…

Edge Decompositions of Hypercubes by Paths and by Cycles

If $H$ is (or is isomorphic to) a subgraph of $G$, $H$ is said to {\it divide} $G$ if there is an edge-decomposition of $G$ by copies of $E(H)$, the edge set of $H$. A more restrictive version of this is when there is a subgroup ${\cal H}$ of {\rm Aut} $(G)$, the automorphism group of $G$, such that the copies of $E(H)$ are the translates of $E(H)$ by the elements of ${\cal H}$. In a paper by the second author, this situation was described by saying that $H$, or more precisely $E(H)$, is a {\it fundamental} set for $G$. Many authors have studied the notion of divisibility for various graphs, a…

Edge colorings of graphs without monochromatic stars

In this note, we improve on results of Hoppen, Kohayakawa and Lefmann about the maximum number of edge colorings without monochromatic copies of a star of a fixed size that a graph on $n$ vertices may admit. Our results rely on an improved application of an entropy inequality of Shearer.

Multicolor Turán numbers

We consider a natural generalisation of Turán's forbidden subgraph problem and the Ruzsa-Szemerédi problem by studying the maximum number $ex_F(n,G)$ of edge-disjoint copies of a fixed graph $F$ can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. We determine the pairs $\{F, G\}$ for which the order of magnitude of $ex_F(n,G)$ is quadratic and prove several asymptotic results using various tools from the regularity lemma and supersaturation to graph packing results.

Multi-Switch: a Tool for Finding Potential Edge-Disjoint $1$-factors

Let $n$ be even, let $π= (d_1, \ldots, d_n)$ be a graphic degree sequence, and let $π- k = (d_1 - k, \ldots, d_n - k)$ also be graphic. Kundu proved that $π$ has a realization $G$ containing a $k$-factor, or $k$-regular graph. Another way to state the conclusion of Kundu's theorem is that $π$ \emph{potentially} contains a $k$-factor. Busch, Ferrara, Hartke, Jacobsen, Kaul, and West conjectured that more was true: $π$ potentially contains $k$ edge-disjoint $1$-factors. Along these lines, they proved $π$ would potentially contain edge-disjoint copies of a $(k-2)$-factor and two $1$-factors. We f…

Stability and Ramsey numbers for cycles and wheels

We study the structure of red-blue edge colorings of complete graphs, with no copies of the $n$-cycle $C_n$ in red, and no copies of the $n$-wheel $W_n = C_n \ast K_1$ in blue, for an odd integer $n$. Our first main result is that in any such coloring, deleting at most two vertices we obtain a vertex-partition of $G$ into three sets such that the edges inside the partition classes are red, and edges between partition classes are blue. As a second result, we obtain bounds for the Ramsey numbers of $r(C_{2k+1},W_{2j})$ for $k < j$ integers, which asymptotically confirm the values of $4j+1$, as i…

通过边粘合实现的超饱和

Supersaturation via edge-gluing

该研究证明在温和对称性条件下边粘合保持Erdős-Simonovits强超饱和猜想的可满足性，还证明将满足弱猜想的图沿子森林粘合所得图也满足弱猜想，推进了极值图论中超饱和问题的研究。

On graphs that contain exactly k copies of a subgraph, and a related problem in search theory

We study $\mathrm{exa}_k(n,F)$, the largest number of edges in an $n$-vertex graph $G$ that contains exactly $k$ copies of a given subgraph $F$. The case $k=0$ is the Turán number $\mathrm{ex}(n,F)$ that is among the most studied parameters in extremal graph theory. We show that for any $F$ and $k$, $\mathrm{exa}_k(n,F)=(1+o(1))\mathrm{ex}(n,F))$ and determine the exact values of $\mathrm{exa}_k(n,K_3)$ and $\mathrm{exa}_1(n,K_r)$ for $n$ large enough. We also explore a connection to the following well-known problem in search theory. We are given a graph of order $n$ that consists of an unknow…

非对角Ramsey重数中的Turán着色

Turán Colourings in Off-Diagonal Ramsey Multiplicity

本研究将Fox和Wigderson关于由带临界边的连通非3-可着色图加悬挂边构成的图类的Ramsey重数常数由Turán着色达到的结果，推广到涉及两图红、蓝副本加权和最小化的非对角Ramsey重数常数变体。

Forbidden pairs for equality of edge-connectivity and minimum degree

Let $\mathcal{H}$ be a class of given graphs. A graph $G$ is said to be $\mathcal{H}$-free if $G$ contains no induced copies of $H$ for any $H \in \mathcal{H}$. In this article, we characterize all pairs $\{R,S\}$ of graphs such that every connected $\{R,S\}$-free graph has the same edge-connectivity and minimum degree.

The probability of nonexistence of a subgraph in a moderately sparse random graph

We develop a general procedure that finds recursions for statistics counting isomorphic copies of a graph $G_0$ in the common random graph models ${\cal G}(n,m)$ and ${\cal G}(n,p)$. Our results apply when the average degrees of the random graphs are below the threshold at which each edge is included in a copy of $G_0$. This extends an argument given earlier by the second author for $G_0=K_3$ with a more restricted range of average degree. For all strictly balanced subgraphs $G_0$, our results gives much information on the distribution of the number of copies of $G_0$ that are not in large "cl…

The number of trees in a graph

Let $T$ be a tree with $t$ edges. We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \[2|E| \prod_{v \in V} (d(v) - t + 1)^{\frac{(t-1)d(v)}{2|E|}}.\] Consequently, any $n$-vertex graph of average degree $d$ and minimum degree at least $t$ contains at least $$nd(d-t+1)^{t-1}$$ isomorphic (labeled) copies of $T$. This answers a question of Dellamonica et. al. (where the above statement was proved when $T$ is the path with three edges) while extending an old result of Erd\H os and Simonovits.

Counting substructures I: color critical graphs

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of $F$ in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits, who proved that there is one copy of $F$, and of Rademacher, Erd\H os and Lovász-Simonovits, who proved similar counting results when $F$ is a complete graph. One of the simplest cases of our theorem is the following new result. There is an absolute positive constant $c$ such that if $n$ is sufficiently large and $1 \le q < cn$, then every $n$ vertex graph wi…

Ringel's tree packing conjecture in quasirandom graphs

We prove that any quasirandom graph with $n$ vertices and $rn$ edges can be decomposed into $n$ copies of any fixed tree with $r$ edges. The case of decomposing a complete graph establishes a conjecture of Ringel from 1963.

Multidesigns for a graph pair of order 6

Given two graphs $G$ and $H$, a $(G,H)$-multidecomposition of $K_{n}$ is a partition of the edges of $K_{n}$ into copies of $G$ and $H$ such that at least one copy of each is used. We give necessary and sufficient conditions for the existence of $(C_{6},\overline{C}_{6})$-multidecomposition of $K_{n}$ where $C_{6}$ denotes a cycle of length 6 and $\overline{C}_{6}$ denotes the complement of $C_{6}$. A $(G,H)$-multipacking of $K_{n}$ is a partition of a subset of the edges of $K_{n}$ into copies of $G$ and $H$ such that at least one copy of each is used. The set consisting of the edges of $K_{n…

On induced Ramsey numbers for multiple copies of graphs

We say that a graph F strongly arrows a pair of graphs (G,H) if any colouring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F. The induced Ramsey number, IR(G,H) is defined as the smallest order of a graph that strongly arrows (G,H). We consider the connection between the induced Ramsey number for a pair of two connected graphs IR(G,H) and the induced Ramsey number for multiple copies of these graphs IR(sG,tH), where xG denotes the pairwise vertex-disjoint union of x copies of G. It is easy to see that if F strongly arrow (G,H), then (s+t-…

Zero-sum copies of spanning forests in zero-sum complete graphs

For a complete graph $K_n$ of order $n$, an edge-labeling $c:E(K_n)\to \{ -1,1\}$ satisfying $c(E(K_n))=0$, and a spanning forest $F$ of $K_n$, we consider the problem to minimize $|c(E(F'))|$ over all isomorphic copies $F'$ of $F$ in $K_n$. In particular, we ask under which additional conditions there is a zero-sum copy, that is, a copy $F'$ of $F$ with $c(E(F'))=0$. We show that there is always a copy $F'$ of $F$ with $|c(E(F'))|\leq Δ(F)+1$, where $Δ(F)$ is the maximum degree of $F$. We conjecture that this bound can be improved to $|c(E(F'))|\leq (Δ(F)-1)/2$ and verify this for $F$ being t…

Linear algebra and bootstrap percolation

In $\HH$-bootstrap percolation, a set $A \subset V(\HH)$ of initially 'infected' vertices spreads by infecting vertices which are the only uninfected vertex in an edge of the hypergraph $\HH$. A particular case of this is the $H$-bootstrap process, in which $\HH$ encodes copies of $H$ in a graph $G$. We find the minimum size of a set $A$ that leads to complete infection when $G$ and $H$ are powers of complete graphs and $\HH$ encodes induced copies of $H$ in $G$. The proof uses linear algebra, a technique that is new in bootstrap percolation, although standard in the study of weakly saturated …

Tight Ramsey bounds for multiple copies of a graph

The Ramsey number $r(G)$ of a graph $G$ is the smallest integer $n$ such that any $2$ colouring of the edges of a clique on $n$ vertices contains a monochromatic copy of $G$. Determining the Ramsey number of $G$ is a central problem of Ramsey theory with long and illustrious history. Despite this there are precious few classes of graphs $G$ for which the value of $r(G)$ is known exactly. One such family consists of large vertex disjoint unions of a fixed graph $H$, we denote such a graph, consisting of $n$ copies of $H$ by $nH$. This classical result was proved by Burr, Erdős and Spencer in 19…

$K_8$ 的 Erdős-Gyárfás 问题的一个变体

A variant of the Erdős-Gyárfás problem for $K_8$

本文构造了一种使用 $n^{o(1)}$ 种颜色且避免 $K_8$ 偶着色副本的边着色，解决了该 Erdős-Gyárfás 问题变体的最小未解情形，并提出了更强的着色条件猜想，同时改进了 $K_4$ 和 $K_5$ 的构造。

在不含\(C_{2\ell + 1}\)的图中的强子图计数稳定性

Strong Subgraph-Count Stability in $C_{2\ell+1}$-Free Graphs

研究不含\(C_{2\ell + 1}\)的图的稳定性问题，通过子图计数展开。对固定的\(\ell\)和\(r\)，若此类图含特定数量的\(P_t\)或\(C_{2a}\)，则有对应结构，还证明了近完全二分图的计数定理。

Triangle-free Subgraphs at the Triangle-Free Process

We consider the triangle-free process: given an integer n, start by taking a uniformly random ordering of the edges of the complete n-vertex graph K_n. Then, traverse the ordered edges and add each traversed edge to an (initially empty) evolving graph - unless its addition creates a triangle. We study the evolving graph at around the time where Θ(n^{3/2 + ε}) edges have been traversed for any fixed ε\in (0,10^{-10}). At that time and for any fixed triangle-free graph F, we give an asymptotically tight estimation of the expected number of copies of F in the evolving graph. For F that is balance…

Ramsey numbers for multiple copies of sparse graphs

For a graph $H$ and an integer $n$, we let $nH$ denote the disjoint union of $n$ copies of $H$. In 1975, Burr, Erdős, and Spencer initiated the study of Ramsey numbers for $nH$, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant $c = c(H)$ such that $r(nH) = (2|H| - α(H))n + c$, provided $n$ is sufficiently large. Subsequently, Burr gave an implicit way of computing $c$ and noted that this long term behaviour occurs when $n$ is triply exponential in $|H|$. Very recently, Bucić and Sudakov revived the problem and established an essentiall…

Decomposing highly edge-connected graphs into paths of any given length

In 2006, Barát and Thomassen posed the following conjecture: for each tree $T$, there exists a natural number $k_T$ such that, if $G$ is a $k_T$-edge-connected graph and $|E(G)|$ is divisible by $|E(T)|$, then $G$ admits a decomposition into copies of $T$. This conjecture was verified for stars, some bistars, paths of length $3$, $5$, and $2^r$ for every positive integer $r$. We prove that this conjecture holds for paths of any fixed length.

$A_α$-spectrum of a graph obtained by copies of a rooted graph and applications

Given a connected graph $R$ on $r$ vertices and a rooted graph $H,$ let $R\{H\}$ be the graph obtained from $r$ copies of $H$ and the graph $R$ by identifying the root of the $i-th$ copy of $H$ with the $i-th$ vertex of $R$. Let $0\leqα\leq1,$ and let \[ A_α(G)=αD(G)+(1-α)A(G) \] where $D(G)$ and $A(G)$ are the diagonal matrix of the vertex degrees of $G$ and the adjacency matrix of $G$, respectively. A basic result on the $A_α-$ spectrum of $R\{H\}$ is obtained. This result is used to prove that if $H=B_{k}$ is a generalized Bethe tree on $k$ levels, then the eigenvalues of $A_α(R\{B_{k}\})$ …

Many $T$ copies in $H$-free graphs

For two graphs $T$ and $H$ with no isolated vertices and for an integer $n$, let $ex(n,T,H)$ denote the maximum possible number of copies of $T$ in an $H$-free graph on $n$ vertices. The study of this function when $T=K_2$ is a single edge is the main subject of extremal graph theory. In the present paper we investigate the general function, focusing on the cases of triangles, complete graphs, complete bipartite graphs and trees. These cases reveal several interesting phenomena. Three representative results are: (i) $ex(n,K_3,C_5) \leq (1+o(1)) \frac{\sqrt 3}{2} n^{3/2},$ (ii) For any fixed $m…

三部染色临界图的边谱过饱和

Edge-spectral supersaturation for tripartite color-critical graphs

该研究针对色数为3的两类染色临界图，证明超过紧谱Turán阈值会迫使出现最优多项式数量的对应图副本，扩展了边谱存在性定理至三色过饱和结果。

Quartic graphs with every edge in a triangle

We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from these by a number of simple subgraph-replacement operations. A corollary of this is that a simple quartic graph with every edge in a triangle is either the square of a cycle, the line graph of a cubic graph or a graph obtained from the line multigraph of a cubic multigraph by replacing triangles with copies of K_{1,1,3}.

Tiling edge-coloured graphs with few monochromatic bounded-degree graphs

We prove that for all integers $Δ,r \geq 2$, there is a constant $C = C(Δ,r) >0$ such that the following is true for every sequence $\mathcal{F} = \{F_1, F_2, \ldots\}$ of graphs with $v(F_n) = n$ and $Δ(F_n) \leq Δ$, for each $n \in \mathbb{N}$. In every $r$-edge-coloured $K_n$, there is a collection of at most $C$ monochromatic copies from $\mathcal{F}$ whose vertex-sets partition $V(K_n)$. This makes progress on a conjecture of Grinshpun and Sárközy.

Ramsey图中不可避免的子图

Unavoidable subgraphs in Ramsey graphs

本文基于围长Ramsey定理给出的“$F$副本森林”结构，推导进一步结论，并刻画这类子图在Ramsey图中必然出现的程度。

Fractional edge-decompositions of dense graphs and related eigenvalues

We consider the problem of decomposing some $t$-uniform hypergraph $G$ into copies of another, say $H$, with nonnegative rational weights. For fixed $H$ on $k$ vertices, we show that this is always possible for all $G$ having sufficiently many vertices and `local density' at least $1-C(t)k^{-2t}$. In the case $t=2$ and $H=K_3$, we show that all large graphs with density at least $27/28$ admit a fractional triangle decomposition. The proof relies on estimates of certain eigenvalues in the Johnson scheme.

子图计数的过饱和

Supersaturation for subgraph counts

本文研究了子图计数的过饱和问题，在 Turán 极值数和 Alon-Shikhelman 广义极值问题的基础上，确定了部分广义极值数并证明了相应的过饱和定理。