Edge number critical triangle free graphs with low independence numbers
Jörgen Backelin
确定了所有无三角形图G =（V，E）的结构，其中|E| - 6 |V| + 13α（G）= 0，从而肯定地回答了StanismartawRadziszowski和DonaldKreher的一个问题.
The structure of all triangle free graphs G = (V,E) with |E| - 6|V| + 13α(G) = 0 is determined, yielding an affirmative answer to a question of Stanisław Radziszowski and Donald Kreher.
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The independence number $α(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free graphs (J. Comb. Theory, Ser. B 53 (1991) 300-307), and considerably improving recent results of Li and Zang (SIAM J. Discrete Math. 20 (2006) 96-104) and Chishti et al. (Acta Univ. Sapientiae, Informatica 6 (2014) 132-158), we show that $$α(H)\geq \sum_{u\in V(H)}f_r(d_H(u))$$ for an $r$-uniform linear triangle-free hypergraph $H$ with $r\geq 2$, where \begin{eqna…
The celebrated Hadwiger's conjecture states that if a graph contains no $K_{t+1}$ minor then it is $t$-colourable. If true, it would in particular imply that every $n$-vertex $K_{t+1}$-minor-free graph has an independent set of size at least $n/t$. In 1982, Duchet and Meyniel proved that this bound holds within a factor $2$. Their bound has been improved; most notably in an absolute factor by Fox, which was later improved by Balogh and Kostochka. Here we consider the same question for triangle-free graphs. By the results of Shearer and Kostochka and Thomason, it follows that any triangle-free …
Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least n/3. We prove that unless the graph contains a certain obstruction, its independence number is at least n/(3-epsilon) for some fixed epsilon>0. We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-free graph G' such that alpha(G')-|G'|/3=alpha(G)-|G|/3 and |G'|<=(alpha(G)-|G|/3)/epsilon. We derive a number of algorithmic consequences as well as a structural description of n-vertex plane tri…
Sharp asymptotics for triangle independence and covering numbers
该研究确定了图论中三角形独立数与覆盖数之和的渐近常数，证明其极限值为$3/2$，解决了Erdős等人提出的相关渐近常数问题。
Steinberg and Tovey proved that every n-vertex planar triangle-free graph has an independent set of size at least (n+1)/3, and described an infinite class of tight examples. We show that all n-vertex planar triangle-free graphs except for this one infinite class have independent sets of size at least (n+2)/3.
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give new bounds on the number of independent sets in triangle-free regular graphs.
We consider the following problem posed by Erdos in 1962. Suppose that $G$ is an $n$-vertex graph where the number of $s$-cliques in $G$ is $t$. How small can the independence number of $G$ be? Our main result suggests that for fixed $s$, the smallest possible independence number undergoes a transition at $t=n^{s/2+o(1)}$. In the case of triangles ($s=3$) we obtain the following result which is sharp apart from constant factors and generalizes basic results in Ramsey theory: there exists $c>0$ such that every $n$-vertex graph with $t$ triangles has independence number at least $$c \cdot \min\l…
We study the number of edges, $e(G)$, in triangle-free graphs with a prescribed number of vertices, $n(G)$, independence number, $α(G)$, and number of cycles of length four, $\operatorname{N}(C_4;G)$. We in particular show that $$3e(G) - 17n(G) + 35α(G) + \operatorname{N}(C_4;G) \geq 0$$ for all triangle-free graphs $G$. We also characterise the graphs that satisfy this inequality with equality.
The independence ratio of 4-cycle-free planar graphs
研究证明无三角形与4圈相邻的平面图的独立比上限为4-ε，ε=1/30，并推广到无4圈或无相邻三角形的平面图，后者进一步提高ε至2/9。
We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly triangle divisible graph that is also triangle decomposable. We explore several classes of planar graphs as well as a class of toroidal graphs.
We show the quarter of a century old conjecture that every $K_4$-free graph with $n$ vertices and $\lfloor n^2/4 \rfloor +k$ edges contains $k$ pairwise edge disjoint triangles.
We investigate the independence number of two graphs constructed from a polarity of $\mathrm{PG}(2,q)$. For the first graph under consideration, the Erdős-Rényi graph $ER_q$, we provide an improvement on the known lower bounds on its independence number. In the second part of the paper we consider the Erdős-Rényi hypergraph of triangles $\mathcal{H}_q$. We determine the exact magnitude of the independence number of $\mathcal{H}_q$, $q$ even. This solves a problem posed by Mubayi and Williford.
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every edge-colored complete graph without containing monochromatic triangles always contains a properly colored Hamilton path. In this paper, we investigate the existence of properly colored cycles in edge-colored complete graphs when monochromatic triangles are forbidden. We obtain a vertex-pancyclic analogous result combined with a characterization of all the excepti…
This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976.
We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an $n$-vertex graph $G$ with sublinear independence number. In this setting, we show that if $δ(G) \ge n/3 + o(n)$ then $G$ has a triangle-tiling covering all but at most four vertices. Also, for every $r \ge 5$, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that $G$ is $K_r$-free and $n$ is divisible by $3$.
Ramsey-Turán Type Problem for Perfect Transitive Triangle Tilings in Digraphs
研究有向图在独立数较小条件下，通过最小度条件保证完美传递三角形铺砌的存在性，给出了渐近最优的最小度下界。
Minimal Regular graphs with every edge in a triangle
本文研究每条边都在三角形中的正则图，证明三角形数量的下界，并对 r≤5 的极小情形利用线图和偶圈分解完成分类，同时给出 r≥6 的构造方法。
An edge dominating set $F$ of a graph $G=(V,E)$ is an \textit{edge cut dominating set} if the subgraph $\langle V,G-F \rangle$ is disconnected. The \textit{edge cut domination number} $γ_{ct}(G)$ of $G$ is the minimum cardinality of an edge cut dominating set of $G.$ In this paper we study the edge cut domination number and investigate its relationships with other parameters of graphs. We also introduce the properties edge cut irredundance and edge cut independence.
We characterise the quintic (i.e. 5-regular) multigraphs with the property that every edge lies in a triangle. Such a graph is either from a set of small graphs or is formed by adding a perfect matching to a line graph of a cubic graph as double edges, or can be reduced by a sequence of operations to one of these graphs.
Quartic graphs with every edge in a triangle
我们证明了四次（即4-正则）重图的每条边都在三角形内的性质。主要的结果是，这样的图要么是平方的周期，线重图的立方重图，或从这些由一些简单的子图替换操作。这一点的一个推论是，每条边都在三角形中的简单四次图要么是圈的平方，要么是三次图的线图，要么是通过用K_{1,1,3}的副本替换三角形而从三次多重图的线图获得的图。
Dense graphs with a large triangle cover have a large triangle packing
众所周知，具有$m$边的图可以通过移除（略小于）$m/2$边而成为无三角形的。另一方面，有许多类的图是很难使三角形自由的意义上说，它是必要的，以消除所有的三角形删除大致$m/2$边。证明了很难使之成为无三角形的稠密图具有大量的两两边不相交三角形。特别地，它们具有多于$m(1/4+cβ^2)$的成对边不相交三角形，其中$β$是图的密度，而$c$是绝对常数。这改善了以前的$m(1/4-o(1))$界，其中遵循的渐近有效性的Tuza的猜想稠密图。证明了这类图有一个大小为$m(1/3-o(1))$的渐近最优三角形填充.这个结果被推广到更大的集团和奇数圈。
The Fractional Chromatic Number of Triangle-free Graphs with $Δ\leq 3$
设$G$是任意一个无三角形的最大度图$Δ\leq 3$。Staton证明了$G$的独立数至少为5/14 n。Heckman和Thomas证明了Staton的结果可以加强为$G$的分数色数的一个界，即$χ_f(G)\leq 14/5. Recently, Hatami and Zhu proved $ χ f（G）\leq 3 -{3/64} $. In this paper, we prove $ χ f（G）\leq 3- 3/43$.
A note on edge colorings distinguishing all triangles in a graph
本文研究图中区分所有三角形的边着色问题，作为区分极大星思想的扩展，估计了正常和一般着色下关于三角形的新不变量。
Rainbow triangles in edge-colored graphs
设$G$是边有色图. $G$的顶点$v$的颜色度被定义为与$v$关联的边的颜色的数量。$G$的颜色数定义为$G$中边的颜色数。彩虹三角形是每对边都有不同颜色的三角形。本文从色度、色数和边数三个方面给出了边有色图中彩虹三角形存在的充分条件。作为推论，李和王提出的一个猜想（边有色图中的着色度和异色圈，European J. Combin. 33（2012）1958- 1964）得到证实。
Maximum edges possible in a graph for restricted independence number, maximum degree, and maximum matching size
本文给出了单图在最大匹配大小、独立数和最大度三个参数中的两个参数为限制值时可能的最大边数的精确界。我们还构造极值图，实现在所有情况下的边缘界限。我们进一步建立这些极值图的唯一性，只要他们是唯一的。
On a problem of Erdős and Rothschild on edges in triangles
埃尔德斯和罗斯柴尔德要求估计最大数目，记为H（N，C），使得每个至少有CN ^2条边的N-顶点图，其中每条边至少包含在一个三角形中，必须包含至少H（N，C）个三角形中的一条边。1987年，埃尔德什特别提出了一个问题：对于所有的C>0，是否存在ε>0，使得H（N，C）> N^ε。我们证明了对于每个固定的C < 1/4，H（N，C）= N^{O（1/log log N）}。这给出了Erdos问题的否定答案，并且在C的范围方面是最好的可能，因为已知每个具有超过（N^2）/4条边的N-顶点图包含至少N/6个三角形中的边。
No Dense Subgraphs Appear in the Triangle-free Graph Process
考虑无三角形图过程，它从$n$顶点上的空图和可能的${n \choose 2}$边的随机排序开始;如果图保持无三角形，则以这种顺序添加边。我们将证明存在一个常数$c$使得在$k$顶点上的任何固定的有限无三角形图的副本至少有$ck$边渐近几乎必然出现在无三角形图过程中。
The edge clique cover number $ecc(G)$ of a graph $G$ is the size of the smallest set of complete subgraphs whose union covers all edges of $G$. It has been conjectured that all the simple graphs with independence number two satisfy $ecc(G)\leq n$. First, we present a class of graphs containing edges difficult to cover but that satisfy the conjecture. Second, we describe a large class of graphs $G$ such that $ecc(G)\leq \frac{3}{2}n$. This class is easy to characterize.
Well-edge-dominated graphs containing triangles
本文完整刻画了恰好含一个三角形的良边支配图（两个无限族），并证明仅有八个良边支配外平面图，多数至多含一个三角形。
Subcubic triangle-free graphs have fractional chromatic number at most 14/5
证明了每一个次立方无三角形图的分数色数至多为14/5，从而证实了Heckman和Thomas的一个猜想[无三角形立方图的独立比的一个新证明.离散数学233（2001），233- 237]。
A note on transformations of edge colorings of chordless graphs and triangle-free graphs
本文指出通过对Bonamy等人（2023）关于无三角形图边着色Kempe变换结论的证明进行微小修改，可将该结论推广到包含无三角形图和无弦图的更大类简单图中。
Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$. We conjecture a natural generalization to partitions with more blocks.
A well-known conjecture by Erdős states that every triangle-free graph on $n$ vertices can be made bipartite by removing at most $n^2/25$ edges. This conjecture was known for graphs with edge density at least $0.4$ and edge density at most $0.172$. Here, we will extend the edge density for which this conjecture is true; we prove the conjecture for graphs with edge density at most $0.2486$ and for graphs with edge density at least $0.3197$. Further, we prove that every triangle-free graph can be made bipartite by removing at most $n^2/23.5$ edges improving the previously best bound of $n^2/18$.
For positive integers $n$ and $r$, we consider $n$-vertex graphs with the maximum number of $r$-edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if $2 \leq r \leq 26$ and $n$ is sufficiently large, the maximum is attained by the bipartite Turán graph $T_2(n)$ on $n$ vertices. This is best possible, as $T_2(n)$ is not extremal for $r \geq 27$ colors and $n \geq 3$.
Let $G$ be a graph of order $n$ with an edge-coloring $c$, and let $δ^c(G)$ denote the minimum color-degree of $G$. A subgraph $F$ of $G$ is called rainbow if any two edges of $F$ have distinct colors. There have been a lot results in the existing literature on rainbow triangles in edge-colored complete graphs. Fujita and Magnant showed that for an edge-colored complete graph $G$ of order $n$, if $δ^c(G)\geq \frac{n+1}{2}$, then every vertex of $G$ is contained in a rainbow triangle. In this paper, we show that if $δ^c(G)\geq \frac{n+k}{2}$, then every vertex of $G$ is contained in at least $k…
For an edge-colored graph, a subgraph is called rainbow if all its edges have distinct colors. We show that if $G$ is an edge-colored graph of order $n$ and size $m$ using $c$ colors on its edges, and $m+c\geq \binom{n+1}{2}+k-1$ for a non-negative integer $k$, then $G$ contains at least $k$ rainbow triangles. For $n\geq 3k$, we show that this result is best possible, and we completely characterize the class of edge-colored graphs for which this result is sharp. Furthermore, we show that an edge-colored graph $G$ contains at least $k$ rainbow triangles if $\sum\limits_{v\in V(G)} d^c_G(v)\geq …
The triangle-free process begins with an empty graph on n vertices and iteratively adds edges chosen uniformly at random subject to the constraint that no triangle is formed. We determine the asymptotic number of edges in the maximal triangle-free graph at which the triangle-free process terminates. We also bound the independence number of this graph, which gives an improved lower bound on the Ramsey numbers R(3,t): we show R(3,t) > (1-o(1)) t^2 / (4 log t), which is within a 4+o(1) factor of the best known upper bound. Our improvement on previous analyses of this process exploits the self-cor…
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 multigraph G is triangle decomposable if its edge set can be partitioned into subsets, each of which induces a triangle of G, and rationally triangle decomposable if its triangles can be assigned rational weights such that for each edge e of G, the sum of the weights of the triangles that contain e equals 1. We present a necessary and sufficient condition for a planar multigraph to be triangle decomposable. We also show that if a simple planar graph is rationally triangle decomposable, then it has such a decomposition using only weights 0,1 and 1/2. This result provides a characterization of…
Independence and Matching Number in Graphs with Maximum Degree 4
我们证明了对于每个最大度至多为$4$的无三角形图$G$，有$\frac{7}{4}α(G)+β(G)\geq n(G)$和$α(G)+\frac{3}{2}β(G)\geq n(G)$，其中$α(G)$是独立数，$β(G)$是$G$的匹配数.这些结果对于$13$顶点上的图是尖锐的。进一步证明了无$\{3K_1,K_1\cup K_5\}$图的$χ(G)\leq \frac{7}{4}ω(G)$，其中$χ(G)$是图的色数，$ω(G)$是$G$的团数.
In an edge-colored graph $(G,c)$, let $d^c(v)$ denote the number of colors on the edges incident with a vertex $v$ of $G$ and $δ^c(G)$ denote the minimum value of $d^c(v)$ over all vertices $v\in V(G)$. A cycle of $(G,c)$ is called proper if any two adjacent edges of the cycle have distinct colors. An edge-colored graph $(G,c)$ on $n\geq 3$ vertices is called properly vertex-pancyclic if each vertex of $(G,c)$ is contained in a proper cycle of length $\ell$ for every $\ell$ with $3 \le \ell \le n$. Fujita and Magnant conjectured that every edge-colored complete graph on $n\geq 3$ vertices with…
Triangle-free Subgraphs at the Triangle-Free Process
我们考虑无三角过程：给定一个整数n，从对完全n-顶点图K_n的边取一致随机序开始。然后，遍历有序的边，并将每个遍历的边添加到一个（最初为空的）演化图中-除非它的添加创建了一个三角形。我们研究了当任意固定ε \in（0，10 ^{-10}）的Θ（n^{3/2 + ε}）条边被遍历时的演化图.此时，对于任意固定的无三角形图F，我们给出了F在演化图中的期望拷贝数的渐近紧估计。对于平衡且密度小于2的F（例如，对于长度至少为4的圈F），我们的论证也给出了演化图中F的拷贝数的紧集中结果。我们的分析结合了Spencer的原始分支过程的方法来分析三角形的自由过程和半随机方法。
The minimal density of triangles in tripartite graphs
在给定边密度的三部图中，确定了三角形的最小密度。这扩展了以前的结果邦迪，沈，Gillassé和Gillassen特征的边密度保证存在的三角形在一个三分图。为了更准确地说，我们表明，适当的加权副本的图形形成删除某个9-圈从$K_{3,3,3}$具有最小的三角形密度之间的所有加权三部图与规定的边缘密度。
Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75 n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely 1. We show that for any graph on…
The triangle-free process
考虑下面的随机图过程。我们从n个顶点上的空图开始，每次添加一条边，其中每条边都是从添加到图中时不形成三角形的潜在边的集合中随机均匀选择的。该过程在最大无串图处终止。在这里，我们分析三角形的过程中，确定可能的数量级的数量级的边缘在最终的图形。作为推论，我们证明了无三角过程很可能产生Ramsey R（3，t）图，也就是说，我们对无三角过程的分析给出了Jeong Han Kim先前建立的R（3，t）上的下界的新证明.所介绍的技术扩展到K_4-free过程，从而建立一个小的改进，最有名的下界的Ramsey数R（4，t）。
We consider incomplete tilings of the equilateral triangle of edge length n that is subdivided into n^2 regular equilateral smaller unit triangles. Pairs of the unit triangles that share a side may be converted into lozenges, leaving some subset of the unit triangles untouched. We count numerically these coverings by lozenges and unit triangles for edge lengths n <= 6: the total and the detailed refinement as a function of the number of lozenges.
The independence number $α(G)$ and the dissociation number ${\rm diss}(G)$ of a graph $G$ are the largest orders of induced subgraphs of $G$ of maximum degree at most $0$ and at most $1$, respectively. We consider possible improvements of the obvious inequality $2α(G)\geq {\rm diss}(G)$. For connected cubic graphs $G$ distinct from $K_4$, we show $5α(G)\geq 3{\rm diss}(G)$, and describe the rich and interesting structure of the extremal graphs in detail. For bipartite graphs, and, more generally, triangle-free graphs, we also obtain improvements. For subcubic graphs though, the inequality cann…
A graph $G$ is called a $(3,j;n)$-minimal Ramsey graph if it has the least amount of edges, $e(3,j;n)$, given that $G$ is triangle-free, the independence number $α(G) < j$ and that $G$ has $n$ vertices. Triangle-free graphs $G$ with $α(G) < j$ and where $e(G) - e(3,j;n)$ is small are said to be almost minimal Ramsey graphs. We look at a construction of some almost minimal Ramsey graphs, called $H_{13}$-patterned graphs. We make computer calculations of the number of almost minimal Ramsey triangle-free graphs that are $H_{13}$-patterned. The results of these calculations indicate that many of t…
We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$ edges has a fractional triangle decomposition, for $n \ge 7$. This is a key ingredient in our proof, given in a companion paper, that every $n$-vertex $2$-coloured complete graph contains $n^2/12 + o(n^2)$ edge-disjoint monochromatic triangles, which confirms a conjecture of Erdős.
On Triangles in Colored Pseudoline Arrangements
研究双色伪直线排列中是否存在双色三角形，并证明了相关超图的独立数上限。
The $1$-nearly edge independence number of a graph
该研究聚焦图的1-近乎边独立数$α_1'(G)$，推导了给定顶点数下图的该参数紧上下界，刻画了对应最小参数值的一般图与连通图，并提出后续研究的开放问题。
We study extremal type problem arising from the question: What is the maximum number of edge-disjoint non-crossing perfect matchings on a set S of 2n points in the plane such that their union is a triangle-free geometric graph? We approach this problem by considering four different situations of S. In particular, in the general position, we obtain (i) a sufficient condition for the existence of n edge-disjoint non-crossing perfect matchings in the general position whose union is a maximal triangle-free geometric graph, and (ii) a lower bound on the number of edge-disjoint non-crossing perfect …
In this paper we study the flow-property of graphs containing a spanning triangle-tree. Our main results provide a structure characterization of graphs with a spanning triangle-tree admitting a nowhere-zero $3$-flow. All these graphs without nowhere-zero $3$-flows are constructed from $K_4$ by a so-called bull-growing operation. This generalizes a result of Fan et al. in 2008 on triangularly-connected graphs and particularly shows that every $4$-edge-connected graph with a spanning triangle-tree has a nowhere-zero $3$-flow. A well-known classical theorem of Jaeger in 1979 shows that every grap…
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.
Kostochka and Yancey proved that every 5-critical graph G satisfies: |E(G)|>= (9/4)|V(G)| - 5/4. A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists e,d > 0 such that if G is a 5-critical graph, then |E(G)| >= (9/4 + e)|V(G)|- 5/4 - dT(G), where T(G) is the maximum number of vertex-disjoint cliques of size three or four where cliques of size four have twice the weight of a clique of size three. As a corollary, a triangle-free 5-critical graph G satisfies: |E(G)|>=(9/4 + e)|V(G)| - 5/4.
We study some amalgamation classes introduced by Cherlin and prove the simplicity of the automorphism groups of the Fra{\"ı}ss{é} limits of these classes. We employ the machinery of stationary independence relations used by Tent and Ziegler.
A celebrated result of Mantel shows that every graph on $n$ vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erdős, says that any such graph must have an edge which is contained in at least $n/6$ triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book numbe…
Counting rainbow triangles in edge-colored graphs
本文研究边染色图中彩虹三角形的计数问题，给出了通过给定顶点的彩虹三角形数量的下界，并证明了在最小颜色度条件下存在 $\Omega(n^2)$ 或 $\Omega(n^3)$ 个彩虹三角形，结果最优且推广了相关定理。
Four triangle-free intrinsically knotted graphs with 22 edges
本文提出基于删除顶点对的诱导子图检测方法，分类了22条边、八个4度顶点和四个3度顶点的无三角形内蕴纽结图，证明恰好有四个。
An improved upper bound for Tuza's conjecture via 2-colorable triangle families
本文针对Tuza猜想，通过研究二染色三角形族的关键性质，将其上界从66/22 ν(G)改进为63/22 ν(G)。
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