Daniel W. Cranston, Sogol Jahanbekam, Douglas B. West
We apply the Discharging Method to prove the 1,2,3-Conjecture and the 1,2-Conjecture for graphs with maximum average degree less than 8/3. Stronger results on these conjectures have been proved, but this is the first application of discharging to them, and the structure theorems and reducibility results are of independent interest.
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
Bollobás and Riordan, in their paper "Metrics for sparse graphs," proposed a number of provocative conjectures extending central results of quasirandom graphs and graph limits to sparse graphs. We refute these conjectures by exhibiting a sequence of graphs with convergent normalized subgraph densities (and pseudorandom $C_4$-counts), but with no limit expressible as a kernel.
A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1973, Erdős conjectured that one can find so-called `sparse' Steiner triple systems. Roughly speaking, the aim is to have at most $j-3$ triples on every set of $j$ points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rödl as…
The triangle packing number $ν(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2ν(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random graph $G=G(n,m)$, when $m \le 0.2403n^{3/2}$ or $m\ge 2.1243n^{3/2}$. This is done by analyzing a greedy algorithm for finding large triangle packings in random graphs.
Resolving the Kohayakawa--Kreuter Conjecture for Families
研究了如何将(a,b)-稀疏图分割为两个稀疏子图，证明了Kuperwasser等人关于家庭族的科哈亚卡-克鲁特猜想的猜想。
For a loopless multigraph $G$, the fractional arboricity $Arb(G)$ is the maximum of $\frac{|E(H)|}{|V(H)|-1}$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G)\le k+\frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture was previously proved for $d=k+1$ and for $k=1$ when $d \le 6$. We prove it for all $d$ when $k \le 2$, except for $(k,d)=(2,1)$.
Loebl, Komlós and Sós conjectured that every $n$-vertex graph $G$ with at least $n/2$ vertices of degree at least $k$ contains each tree $T$ of order $k+1$ as a subgraph. We give a sketch of a proof of the approximate version of this conjecture for large values of $k$. For our proof, we use a structural decomposition which can be seen as an analogue of Szemerédi's regularity lemma for possibly very sparse graphs. With this tool, each graph can be decomposed into four parts: a set of vertices of huge degree, regular pairs (in the sense of the regularity lemma), and two other objects each exhibi…
Sparse groups need not be semisparse
本文反驳Hartley关于稀疏群必为半稀疏群的猜想，说明反例源于非忠实maniplex，并证明秩3成立而秩 $n\geq4$ 均存在反例。
One of Erdos's conjectures states that every triangle-free graph on $n$ vertices has an induced subgraph on $n/2$ vertices with at most $n^2/50$ edges. We report several partial results towards this conjecture. In particular, we establish the new bound $\frac{27}{1024}n^2$ on the number of edges in general case. We completely prove the conjecture for graphs of girth $\geq 5$, for graphs with independence number $\geq 2n/5$ and for strongly regular graphs. Each of these three classes includes both known (conjectured) extremal configurations, the 5-cycle and the Petersen graph.
A conjecture of Chung and Graham states that every $K_4$-free graph on $n$ vertices contains a vertex set of size $\lfloor n/2 \rfloor$ that spans at most $n^2/18$ edges. We make the first step toward this conjecture by showing that it holds for all regular graphs.
Erdős conjectured that every triangle-free graph $G$ on $n$ vertices contains a set of $\lfloor n/2 \rfloor$ vertices that spans at most $n^2 /50$ edges. Krivelevich proved the conjecture for graphs with minimum degree at least $\frac{2}{5}n$. Keevash and Sudakov improved this result to graphs with average degree at least $\frac{2}{5}n$. We strengthen these results by showing that the conjecture holds for graphs with minimum degree at least $\frac{5}{14}n$ and for graphs with average degree at least $(\frac{2}{5} - \varepsilon)n$ for some absolute $\varepsilon >0$. Moreover, we show that the c…
Szemerédi's Regularity Lemma is an important tool for analyzing the structure of dense graphs. There are versions of the Regularity Lemma for sparse graphs, but these only apply when the graph satisfies some local density condition. In this paper, we prove a sparse Regularity Lemma that holds for all graphs. More generally, we give a Regularity Lemma that holds for arbitrary real matrices.
On the Holroyd-Talbot Conjecture for Sparse Graphs
本文针对稀疏图研究Holroyd-Talbot的Erdős-Ko-Rado型独立集相交族猜想，分别证明了有界平均度图、有界度图及分裂顶点有界的树在对应r的阶数条件下，存在最大规模的星型独立r-集相交族。
Developing further Stein's recent notion of relative end degrees in infinite graphs, we investigate which degree assumptions can force a locally finite graph to contain a given finite minor, or a finite subgraph of given minimum degree. This is part of a wider project which seeks to develop an extremal theory of sparse infinite graphs.
We study a metric on the set of finite graphs in which two graphs are considered to be similar if they have similar bounded dimensional "factors". We show that limits of convergent graph sequences in this metric can be represented by symmetric Borel measures on $[0,1]^2$. This leads to a generalization of dense graph limit theory to sparse graph sequences.
Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\bf graded sparse graphs}, arising from generically pinned (completely immobilized) bar-and-joint frameworks and prove that they also form matroids. We address five problems on graded sparse graphs: {\bf Decision}, {\bf Extraction}, {\bf Components}, {\bf Optimization}, and {\bf Extension}. We extend our {\bf pebble game algorithms} to solve them.
In the paper, we prove that existential monadic second order convergence law fails for the binomial random graph $G(n,n^{-α})$ for every $α\in(0,1)$.
In a series of four papers we prove the following relaxation of the Loebl-Komlos-Sos Conjecture: For every $α>0$ there exists a number $k_0$ such that for every $k>k_0$ every $n$-vertex graph $G$ with at least $(\frac12+α)n$ vertices of degree at least $(1+α)k$ contains each tree $T$ of order $k$ as a subgraph. The method to prove our result follows a strategy similar to approaches that employ the Szemerédi regularity lemma: we decompose the graph $G$, find a suitable combinatorial structure inside the decomposition, and then embed the tree $T$ into $G$ using this structure. Since for sparse g…
We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order \[ Ω\left( \sqrt{\frac{n d}{\log d}} \right). \] As a corollary we determine the order of a largest complete minor one can guarantee in $d$-regular graphs for which the second largest eigenvalue is bounded away from $d/2$, in $(d/n, o(d))$-jumbled graphs, and in random $d$-regular graphs, for almost all $d = d(n)$.
Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\left\{1, \ldots, n\right\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $\limsup_{n \to \infty} m/n<1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values.
In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings and orthogonal colourings is established. This relation is applied to find independent coverings of some sparse partite graphs. Additionally, a degree condition providing the existence of an independent covering in the case where the graph has a square number of vertices is found.
We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.
We consider the number of vertices that must be removed from a graph G in order that the remaining subgraph has no component with more than k vertices. Our principal observation is that, if G is a sparse random graph or a random regular graph on n vertices with n tending to infinity, then the number in question is essentially the same for all values of k such that k tends to infinity but k=o(n).
Scott conjectured that the class of graphs with no induced subdivision of a given graph is $χ$-bounded. We verify his conjecture for maximal triangle-free graphs.
A counterexample to a basis conjecture of Brualdi, Friedland, and Pothen
通过一个小型稀疏-一般反例，反驳了Brualdi、Friedland和Pothen关于矩阵行空间中初等向量的秩交集猜想。
A {\bf map} is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into $k$ edge-disjoint maps after: (1) the addition of {\it any} $\ell$ edges; (2) the addition of {\it some} $\ell$ edges. These graphs are identified with classes of {\it sparse} graphs; the results are also given in matroidal terms.
In this paper we proof that any cactus graph satisfies graph complement conjecture by finding a orthogonal representation of its complement in $\mathbb{R}^5$.
We prove that there exists a positive constant εsuch that if \log n / n \le p \le n^{-1+ε}, then asymptotically almost surely the random graph G ~ G(n,p) contains a collection of \lfloor δ(G)/2 \rfloor edge-disjoint Hamilton cycles.
We give Henneberg-type constructions for three families of sparse colored graphs arising in the rigidity theory of periodic and other forced symmetric frameworks. The proof method, which works with Laman-sparse finite covers of colored graphs highlights the connection between these sparse colored families and the well-studied matroidal (k, l)-sparse families.
The {\em independence ratio} of a graph $G$ is defined by \[ ι(G) := \sup_{X \subset V(G)} \frac{|X|}{α(X)},\] where $α(X)$ is the independence number of the subgraph of $G$ induced by $X$. The independence ratio is a relaxation of the chromatic number $χ(G)$ in the sense that $χ(G) \geq ι(G)$ for every graph $G$, while for many natural classes of graphs these quantities are almost equal. In this paper, we address two old conjectures of Erdős on cycles in graphs with large chromatic number and a conjecture of Erdős and Hajnal on graphs with infinite chromatic number.
Fullerene graphs, i.e., 3-connected planar cubic graphs with pentagonal and hexagonal faces, are conjectured to be Hamiltonian. This is a special case of a conjecture of Barnette and Goodey, stating that 3-connected planar graphs with faces of size at most 6 are Hamiltonian. We prove the conjecture.
The Ferrers bound conjecture is a natural graph-theoretic extension of the enumeration of spanning trees for Ferrers graphs. We document the current status of the conjecture and provide a further conjecture which implies it.
Characterization of sparse monotone graph classes with bounded domination-to-2-independence ratio
本文研究了具有受限支配数到2-独立数比的稀疏单调图类的特征化问题。
The Turán number of a graph $H$, denoted $\mbox{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph with no subgraph isomorphic to $H$. Solymosi conjectured that if $H$ is any graph and $\mbox{ex}(n,H) = O(n^α)$ where $α> 1$, then any $n$-vertex graph with the property that each edge lies in exactly one copy of $H$ has $o(n^α)$ edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when $H$ is a non-bipartite graph, in particular when $H$ is a triangle, due to Ruzsa and Szemerédi. Using Sidon sets we exhib…
We show that every locally sparse graph contains a linearly sized expanding subgraph. For constants $c_1>c_2>1$, $0<α<1$, a graph $G$ on $n$ vertices is called a $(c_1,c_2,α)$-graph if it has at least $c_1n$ edges, but every vertex subset $W\subset V(G)$ of size $|W|\le αn$ spans less than $c_2|W|$ edges. We prove that every $(c_1,c_2,α)$-graph with bounded degrees contains an induced expander on linearly many vertices. The proof can be made algorithmic. We then discuss several applications of our main result to random graphs, to problems about embedding graph minors, and to positional games.
We study some versions of the statement of Hadwiger's conjecture for finite as well as infinite graphs.
Reed Conjecture is open for more than 20 years now. Here we prove that Reed Conjecture is valid for (1) {P4UnionK1, Kite}-free graphs (2) {Chair, Kite}-free graphs (3) {K2UnionK2complement , H}-free graphs and (4) {2K2, M}-free graphs where H and M are graphs on six vertices each. Reed conjecture is still open in general.
Tutte's $3$-flow conjecture says that every $4$-edge-connected graph admits a nowhere-zero $3$-flow. Kochol (2001) showed that it is enough to prove this conjecture for $5$-edge-connected graphs. Former, Jaeger, Linial, Payan, and Tarsi (1992) conjectured that every $5$-edge-connected graph is $Z_3$-connected and so it admits a nowhere-zero $3$-flow. In this note, we show that if the second conjecture would be true, then every $3$-tree-connected graph must also be $Z_3$-connected and so Tutte's $3$-flow conjecture can be extended to this family of graphs.
Bouchet's conjecture for cyclically 5-edge-connected, cubic signed graphs
该研究证明了布切特猜想在循环五边连通立方符号图中成立，表明此类图具有处处非零的六流。
A conjecture implying Thomassen's chord conjecture in graph theory
本文提出一个蕴含 Thomassen 弦猜想的新猜想，涉及 k-连通图中最长路径的界顶点，并提供支持证据和特例分析，以探索该著名未解问题的周边环境。
A connected $n$-chromatic graph $G$ is double-critical if for all the edges $xy$ of $G$, the graph $G-x-y$ is $(n-2)$-chromatic. In 1966, Erd\H os and Lovász conjectured that the only double-critical $n$-chromatic graph is $K_n$. This conjecture remains unresolved for $n \ge 6.$ In this short note, we verify this conjecture for claw-free graphs $G$ of chromatic number $6$.
The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $Δ$ and diameter $k$. For fixed $k$, the answer is $Θ(Δ^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $Θ(Δ^{k-1})$, and for graphs of bounded arboricity the answer is $Θ(Δ^{\floor{k/2}})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given t…
The dimension of sparse and co-sparse random graph orders
本文研究了稀疏和共稀疏随机图顺序的维度，分析了不同p值下的相变现象，并建立了偏序维度的通用上界。
Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems $(X,Σ)$, where each element $x \in X$ lies in $t$ randomly selected sets of $Σ$, where $t$ is an integer parameter. We provide new bounds in two regimes of parameters. We show that when $|Σ| \ge |X|$ the hereditary discrepancy of $(X,Σ)$ is with high probability $O(\sqrt{t \log t})$; and when $|X| \gg |Σ|^t$ the hereditary discrepancy of $(X,Σ)$ is with high probability $O(1)$. The first bound combines the Lov{á}sz Local Lemma with a new argument based on partial…
The Weak Version of the Graph Complement Conjecture and Partial Results for the Delta Conjecture
本文利用Mader经典结果建立了图补猜想的弱版本，提出了$δ$-猜想的极值解，结合图退化改进了界限，并列出了Colin de Verdière数半正定变体的猜想界限。
Hoffmann-Ostenhof's Conjecture states that states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a $2$-regular subgraph. In this paper, we show that the conjecture holds for claw-free cubic graphs.
In a recent work, Allen, Böttcher, Hàn, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Komlós, Sárközy, and Szemerédi. Roughly speaking, they showed that with high probability in the random graph $G_{n,p}$ for $p \geq C(\log n/n)^{1/Δ}$, sparse regular pairs behave similarly as complete bipartite graphs with respect to embedding a spanning graph $H$ with $Δ(H) \leq Δ$. However, this is typically only optimal when $Δ\in \{2,3\}$ and $H$ either contains a triangle ($Δ= 2$) or many copi…
List coloring $C_3$-free planar graphs with a sparse matching of restricted lists
本文针对不含三角形的平面图，证明了当接收3元列表的顶点集诱导稀疏匹配时，Hu和Zhu提出的列表着色猜想成立，为该猜想提供了新的关键佐证。
The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.
We establish a so-called counting lemma that allows embeddings of certain linear uniform hypergraphs into sparse pseudorandom hypergraphs, generalizing a result for graphs [Embedding graphs with bounded degree in sparse pseudorandom graphs, Israel J. Math. 139 (2004), 93-137]. Applications of our result are presented in the companion paper [Counting results for sparse pseudorandom hypergraphs II].
Borodin & Kostochka conjectured that if maximum degree of a graph is greater than or equal to 9, then the chromatic number of the graph is less than or equal to maximum of ω and maximum degree minus 1. Here we prove that this Conjecture is true for {P3 UNION K1}-free graphs and {K2 UNION complement of K2}-free graphs.
Here we prove that Reed Conjecture is valid for {P5, Flag_Complement}-free graphs where FlagComplement is the complement of the Flag graph. Some of the known results follow as corollaries to our result. Reed conjecture is still open in general.
We show that for any $d=d(n)$ with $d_0(ε) \le d =o(n)$, with high probability, the size of a largest induced cycle in the random graph $G(n,d/n)$ is $(2\pm ε)\frac{n}{d}\log d$. This settles a long-standing open problem in random graph theory.
Let $f_r(n,v,e)$ denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices, in which the union of any $e$ distinct edges contains at least $v+1$ vertices. The study of $f_r(n,v,e)$ was initiated by Brown, Erd{ő}s and S{ó}s more than forty years ago. In the literature, the following conjecture is well known. Conjecture: $n^{k-o(1)}<f_r(n,er-(e-1)k+1,e)=o(n^k)$ holds for all fixed integers $r>k\ge 2$ and $e\ge 3$ as $n\rightarrow\infty$. For $r=3, e=3, k=2$, the bound $n^{2-o(1)}<f_3(n,6,3)=o(n^2)$ was proved by the celebrated (6,3)-theorem of Ruzsa and Szemer{é}di. In this…
Seymour's Second Neighborhood Conjecture (SNC) asserts that every oriented graph has a vertex whose first out-neighborhood is at most as large as its second out-neighborhood. In this paper, we prove that if $G$ is a graph containing no induced $C_4$, $\overline{C_4}$, $S_3$, chair and $\overline{chair}$, then every oriented graph missing $G$ satisfies this conjecture. As a consequence, we deduce that the conjecture holds for every oriented graph missing a threshold graph, a generalized comb or a star.
For any $S\subset [n]$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is a given graph $H$ on the vertex set $S$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{\bf d}$, the probability space of random graphs with a given degree sequence $\bf d$.
The Gamma-Theta Conjecture holds for planar graphs
本文证明了Gamma-Theta猜想对平面图成立，即当平面图的支配数等于永恒支配数时，它也等于团覆盖数。
Sparse $k$-AP Covering Sets and the Arithmetic Kakeya Conjecture
该研究否定了Kiss等人关于k-AP覆盖集的猜想，证明k≥6时存在满足特定密度界的k-AP覆盖集，并将该问题与算术Kakeya猜想建立关联。
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollobás and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize converge…
Sparse String Graphs and Region Intersection Graphs over Minor-Closed Classes have Linear Expansion
研究证明了有限表面中稀疏字符串图具有线性扩张，扩展至任意proper minor-closed类的稀疏区域交图，并应用于图着色。
Let G be a simple graph and L = L(G) the Laplacian matrix of G. G is called L-integral if all its Laplacian eigenvalues are integer numbers. It is known that every cograph, a graph free of P4, is L-integral. The class of P4-sparse graphs and the class of P4-extendible graphs contain the cographs. It seems natural to investigate if the graphs in these classes are still L-integral. In this paper we characterized the L-integral graphs for both cases, P4-sparse graphs and P4-extendible graphs.
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。