Guoli Ding
The purpose of this paper is to characterize graphs that do not have a large $K_{2,n}$-minor. As corollaries, it is proved that, for any given positive integer $n$, every sufficiently large 3-connected graph with minimum degree at least six, every 4-connected graph with a vertex of sufficiently high degree, and every sufficiently large 5-connected graph must have a $K_{2,n}$-minor.
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The Minor Crossing Number of Graphs with an Excluded Minor
图$G$的“次交叉数”是包含$G$作为次交叉的图的最小交叉数。证明了对每个图$H$，都存在一个常数$c$，使得不含$H$ -子图的图$G$最多有$c|V(G)|$的子交叉数.
Framed $4$-valent Graph Minor Theory II: Special Minors and New Examples
本文对框架$4$ -图子式理论进行了研究，它是在“框架$4$ -价图子式理论I”中提出的.介绍。证明了任意框架4-图的平面性定理，并证明了在射影平面中可嵌入性的类似结果。
In this paper, we first introduce the notions of checkerboard colourable minors for ribbon graphs motivated by the Eulerian ribbon graph minors, and two kinds of bipartite minors for ribbon graphs, one of which is the dual of the checkerboard colourable minors and the other is motivated by the bipartite minors of abstract graphs. Then we give an excluded minor characterization of the class of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.
Complete Graph Minors and the Graph Minor Structure Theorem
Robertson和Seymour的图的子结构定理表明，每个排除固定子的图都可以由四个成分的组合构成：嵌入有界亏格的曲面中的图，有界宽度的有界数量的涡，有界数量的顶点，以及cn-sum运算。本文研究了一个逆问题：在用这四个成分构成的图中，一个完全图子图的最大阶是多少？我们的主要结果回答了这个问题的一个常数因子。
A graph $U$ is universal for a graph class $\mathcal{C}\ni U$, if every $G\in \mathcal{C}$ is a minor of $U$. We prove the existence or absence of universal graphs in several natural graph classes, including graphs component-wise embeddable into a surface, and graphs forbidding $K_5$, or $K_{3,3}$, or $K_\infty$ as a minor. We prove the existence of uncountably many minor-closed classes of countable graphs that (do and) do not have a universal element. Some of our results and questions may be of interest to the finite graph theorist. In particular, one of our side-results is that every $K_5$-m…
Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We strengthen this result by showing that the same conclusion holds, if $G$ contains every countable planar graph as a topological minor. In particular, there is no countable planar graph containing every countable planar graph as a topological minor, answering a question by Diestel and Kühn. Moreover, we construct a locally finite planar graph which contains ever…
A graph is said to be \textit{$H$-minor free} if it does not contain $H$ as a minor. In this paper, we characteristic the unique extremal graph with maximal $Q$-index among all $n$-vertex $K_{1,t}$-minor free graphs ($t\ge3$).
Let ${\rm ex \,} {\mathcal B}$ be a minor-closed class of graphs with a set ${\mathcal B}$ of minimal excluded minors. We study (a) the asymptotic number of graphs without $k+1$ disjoint minors in ${\mathcal B}$ and (b) the properties of a uniformly random graph drawn from all such graphs on vertices $\{1,\dots,n\}$. We present new results in the case when ${\rm ex \,} {\mathcal B}$ contains arbitrarily large fans for a general (good enough) set of forbidden minors ${\mathcal B}$. A particular case where our results hold is ${\mathcal B} = \{K_4\}$. For any fixed $k = 1, 2, \dots$ we derive pr…
A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a vertex-minor isomorphic to $F_k$. We also prove that for all positive integers $q$ and $k\ge 3$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a pivot-minor isomorphic to a cycle of length $k$.
Eulerian-minors and a concise recursive characterization of 4-regular planar graphs
本文利用欧拉子式关系刻画欧拉图、平面欧拉图和外平面欧拉图，并给出4-正则平面图的简洁递归刻画。
In the 1960s, Erdős and Pósa proved that there is a packing-covering duality for cycles in graphs. As part of the graph minor project, Robertson and Seymour greatly extended this: there is such a duality for $H$-expansions in graphs if and only if $H$ is a planar graph (this includes the previous result for $H=K_3$). We consider vertex labelled graphs and minors and provide such a characterisation for $2$-connected labelled graphs $H$.
We give an excluded minor characterisation of the class of ribbon graphs that admit partial duals of Euler genus at most one.
Minors in expanding graphs
推广了以前的一些结果，我们得到了几乎紧估计的最大尺寸的cn-minor在各种类别的扩展图。这些结果可以用来证明不含短圈的图和其它H-自由图中含有大的子图，解决了这一领域的一些悬而未决的问题。
Coloration of $K_7^-$-minor free graphs
Hadwiger猜想指出，每个$K_t$ -子自由图都是$(t - 1)$ -可着色的.这个问题已被证明为$t \leq 6$，但仍然开放的$t \geq 7$。$K_7$ -minor free图已被证明是$8$ -可着色的（Albar & Gonçalves，2013）。证明了无$K_7^-$ -子图是$7$ -可着色的，其中$K_7^-$是由$K_7$去掉一条边得到的图.
A short proof of the existence of a minor-universal countable planar graph
本文提出了一种新的、更简短的次泛通用平面图构造方法，证明了其存在性。
In this note, we show that locally finite quasi-transitive graphs are quasi-isometric to trees if and only if every other locally finite quasi-transitive graph quasi-isometric to them is minor excluded. This generalizes results by Ostrovskii and Rosenthal and by Khukhro on minor exclusion for groups.
New minor minimal non-apex graphs
本文研究极点图的极小禁止子图，列举了顶点数≤12和边数≤26的极小禁止子图，并证明了阶数为13的图要么是极点图要么包含K_6子图，验证了Jørgensen猜想。
Hadwiger's conjecture claims that any graph with no $K_t$ minor is $(t - 1)$-colorable. This has been proved for $t \le 6$, but remains open for $t \ge 7$. As a variant of this conjecture, graphs with no $K_t^=$ minor have been considered, where $K_t^=$ denotes the complete graph with two edges removed. It has been shown that graphs with no $K_t^=$ minor are $(2t - 8)$-colorable for $t \in \{7, 8\}$. In this paper, we extend this result to the case $t = 9$ and show that graphs with no $K_9^=$ minor are $10$-colorable.
Motivated by the famous Hadwiger's Conjecture, we study the properties of $8$-contraction-critical graphs with no $K_7$ minor; we prove that every $8$-contraction-critical graph with no $K_7$ minor has at most one vertex of degree $8$, where a graph $G$ is $8$-contraction-critical if $G$ is not $7$-colorable but every proper minor of $G$ is $7$-colorable. This is one step in our effort to prove that every graph with no $K_7$ minor is $7$-colorable, which remains open.
Tutte showed that $4$-connected planar graphs are Hamiltonian, but it is well known that $3$-connected planar graphs need not be Hamiltonian. We show that $K_{2,5}$-minor-free $3$-connected planar graphs are Hamiltonian. This does not extend to $K_{2,5}$-minor-free $3$-connected graphs in general, as shown by the Petersen graph, and does not extend to $K_{2,6}$-minor-free $3$-connected planar graphs, as we show by an infinite family of examples.
A simplified construction is presented for Komjáth's result that for every uncountable cardinal $κ$, there are $2^κ$ graphs of size $κ$ none of them being a minor of another.
This paper considers the following question: What is the maximum number of $k$-cliques in an $n$-vertex graph with no $K_t$-minor? This question generalises the extremal function for $K_t$-minors, which corresponds to the $k=2$ case. The exact answer is given for $t\leq 9$ and all values of $k$. We also determine the maximum total number of cliques in an $n$-vertex graph with no $K_t$-minor for $t\leq 9$. Several observations are made about the case of general $t$.
In 2009, Krivelevich and Sudakov studied the existence of large complete minors in $(t,α)$-expanding graphs whenever the expansion factor $t$ becomes super-constant. In this paper, we give an extension of the results of Krivelevich and Sudakov by investigating a connection between the existence of large complete minors in graphs and good vertex expansion properties.
Forbidden Minors For 3-Connected Graphs With No Non-Splitting 5-Configurations
对于一组五条边，如果相关的道奇森多项式之一等于零，则图分裂。图G的每五条边都可分裂是一个小闭性质。因此，有一个有限集的禁止子F，使得如果一个图H不包含一个子同构于任何图F，那么H分裂。在本文中，我们证明了，如果一个图G是简单的，3-连通的，并分裂，那么G必须不包含任何子同构K5，K3，3，八面体，立方体，或图是一个单一的delta-Y变换远离立方体。因此，这是所有简单3连通禁止子式的集合。2-连通或非简单禁止子式的完整集合仍然没有解决，尽管已经发现了一些。
The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, is minor closed and contains both the class of lifted-graphic matroids and the class of frame matroids, each of which generalises the class of graphic matroids. In this paper, we prove that the matroids $U_{3,7}$ and $U_{4,7}$ are the only $9$-connected excluded minors for the class of quasi-graphic matroids.
Let $V_{8}+e$ denote the unique graph obtained from the Wagner graph, also known as $V_{8}$, by adding an edge between two vertices of distance 3 on the Hamilton cycle, which is exactly a split of a minor of the Petersen graph. A complete characterization of all internally 4-connected graphs with no $V_{8}$ minor is given in [J. Maharry and N. Robertson, The structure of graphs not topologically containing the Wagner graph, J. Combin. Theory Ser. B 121 (2016) 398-420]. In this paper we characterize all internally 4-connected graphs with no $V_{8}+e$ minor.
On triangles in K_r-minor free graphs
本文研究了与小度顶点（分别为7和9）相邻的每条边属于多个三角形（分别为4和5）的图，并证明了这些图包含一个完全图（分别为K_6和K_7）作为子图。第二个案例解决了Nevo问题（Nevo，2007）。如果图的每条边都属于6个三角形，则该图含有一个K_8-子图或含有K_{2,2,2,2,2}作为诱导子图.然后，我们展示了这些结构性质的应用程序，以应力自由度和着色的图形。特别地，在Hadwiger猜想的启发下，我们证明了每个K_7-子自由图是8-可着色的，每个K_8-子自由图是10-可着色的。
Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. This is a strengthening of the famous Hadwiger's Conjecture. Geelen et al. proved that every graph with no odd $K_t$ minor is $O(t\sqrt{\log t})$-colorable. Using the methods the present authors and Postle recently developed for coloring graphs with no $K_t$ minor, we make the first improvement on this bound by showing that every graph with no odd $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> 1/4$.
Forcing large complete minors in infinite graphs
众所周知，在有限图中，可以通过假设大的平均度来强制大的完全子式/拓扑子式。我们的目标是将这一事实扩展到无限图。对于这一点，我们概括的概念，相对结束度，这是以前介绍的第一作者为局部有限图，并表明，大的最小相对度的两端和大的最小程度的顶点意味着存在大的完整（拓扑）未成年人在无限的图可数多个结束。
Interval minors of bipartite graphs were introduced by Jacob Fox in the study of Stanley-Wilf limits. Recently, Mohar, Rafiey, Tayfeh-Rezaie and Wu investigated the maximum number of edges in $K_{k,\ell}$-interval minor free bipartite graphs when $k=2$ and $k=3$. In this paper, we investigate the maximum number of edges in $K_{k,\ell}$-interval minor free bipartite graphs for general $k$ and $\ell$. We also study the maximum number of edges in $K_{\ell_1,\ell_2,\cdots,\ell_t}$-interval minor free multipartite graphs.
In this paper we give structural characterizations of graphs not containing rooted $K_{4}$, $W_{4}$, $K_{2,4}$, and a graph we call $L$.
A mixed graph is obtained from an unoriented graph by orienting a subset of its edges. Yu, Liu, and Qu in 2017 have established the expression for the determinant of Hermitian (quasi-) Laplacian matrix of a mixed graph. Here we find general expressions for all minors of Hermitian (quasi-) Laplacian matrix of mixed graphs.
A graph is sub-unicyclic if it contains at most one cycle. We also say that a graph $G$ is $k$-apex sub-unicyclic if it can become sub-unicyclic by removing $k$ of its vertices. We identify 29 graphs that are the minor-obstructions of the class of $1$-apex sub-unicyclic graphs, i.e., the set of all minor minimal graphs that do not belong in this class. For bigger values of $k$, we give an exact structural characterization of all the cactus graphs that are minor-obstructions of $k$-apex sub-unicyclic graphs and we enumerate them. This implies that, for every $k$, the class of $k$-apex sub-unicy…
On the growth rate of minor-closed classes of graphs
一个子闭图类是一组在同构和取子下闭的标号图。对于一个次闭类$C$，我们设$c_n$为$C$中有$n$顶点的图的个数. Norine等人最近的一个结果表明，对于所有的小闭类$C$，存在一个常数$r$，使得$c_n < r^n n!$。我们的主要结果表明，$c_n$的增长率是远远不是任意的。例如，没有小闭类$C$具有$c_n= r^{n+o(n)} n!$和$0 < r < 1$或$1 < r < ξ\approx 1.76$。
Given a minor-closed class of graphs $\mathcal{G}$, what is the infimum of the non-trivial roots of the chromatic polynomial of $G \in \mathcal{G}$? When $\mathcal{G}$ is the class of all graphs, the answer is known to be $32/27$. We answer this question exactly for three minor-closed classes of graphs. Furthermore, we conjecture precisely when the value is larger than $32/27$.
Every longest circuit of a 3-connected, $K_{3,3}$-minor free graph has a chord
Carsten Schlassen证明了3-连通图中的每一个最长圈都有弦.我们证明了猜想的图没有$K_{3,3}$子，从而为可平面图。
We demonstrate that for every positive integer $Δ$, every K\_4-minor-free graph with maximum degree $Δ$ admits an equitable coloring with k colors wherek $\ge$ ($Δ$+3)/2. This bound is tight and confirms a conjecture by Zhang and Whu. We do not use the discharging method but rather exploit decomposition trees of K 4-minor-free graphs.
Rank-width and Tree-width of H-minor-free Graphs
本文证明了对任意固定的r>=2，不含K_r拓扑子式的图的树宽（分别为K_r = 0，K_r = 0）。作为子图）由线性（分别地，多项式）的函数。我们也给出了其他图类，如K_r-子自由图和有界亏格图的界的改进。
Homological face-width condition forcing $K_6$-minors in graphs on surfaces
证明了嵌入在非分离面宽至少为$7$的（非球面）曲面上的每个图都包含一个同构于$K_6$的子图。对于射影平面上的图，面宽为4也得到同样的结论。
On the number of graphs not containing $K_{3,3}$ as a minor
我们得到精确的渐近估计的标记图的数量不包含$K_{3,3}$作为一个未成年人，也为那些边最大。此外，我们建立了随机$K_{3,3}$ -未成年人自由图的参数的极限定律，如预期的边缘数。为了建立这些结果，我们将相应的图类分解成生成函数的方程，并使用奇异性分析。我们还发现了一个精确的估计不包含图$K_{3,3}$加上一个边作为一个未成年人的图形的数量。
Extremal results regarding $K_6$-minors in graphs of girth at least 5
证明了围长为$\geq 6$的6-连通图都有$K_6$ -子式，从而解决了围长为$ \geq 6$的图的Jorgensen猜想.放松围长的假设，证明了每一个长为$\geq 3 1/5 n-8$和围长为$\geq 5$的6-连通$n$ -顶点图都包含一个$K_6$ -子图.
Universal graphs with forbidden wheel minors
本文证明对任意禁止轮图子式的可数图类，存在一个泛图包含该类中所有图作为诱导子图。
Motivated by Hadwiger's conjecture, we prove that every $n$-vertex graph $G$ with no independent set of size three contains an $\lceil n/2\rceil$-vertex simple minor $H$ with $$0.98688 \cdot \binom{|V(H)|}{2} - o(n^2)$$ edges.
Universal graphs for the topological minor relation
本文证明可数平面图类中不存在拓扑子式万有图，回答了Diestel和Kühn的问题并加强了Pach的结果，同时刻画了细分星图T使得可数T-自由图类具有拓扑子式万有图的条件。
We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.
Minors play an important role in extremal graph theory and spectral extremal graph theory. Tait [The Colin de Verdière parameter, excluded minors, and the spectral radius, J. Combin. Theory Ser. A 166 (2019) 42--58] determined the maximum spectral radius and characterized the unique extremal graph for $K_r$-minor free graphs of sufficiently large order $n$, he also made great progress on $K_{s,t}$-minor free graphs and posed a conjecture: Let $2\leq s\leq t$ and $n-s+1=pt+q$, where $n$ is sufficiently large and $1\leq q\leq t.$ Then $K_{s-1}\nabla (pK_t\cup K_q)$ is the unique extremal graph w…
Forcing finite minors in sparse infinite graphs by large-degree assumptions
进一步发展斯坦最近的概念，在无限图的相对端度，我们调查的程度假设可以迫使一个局部有限图包含一个给定的有限未成年人，或一个有限的子图给定的最低程度。这是一个更广泛的项目的一部分，该项目旨在发展稀疏无限图的极值理论。
Independent Sets in Graphs with an Excluded Clique Minor
设$G$是一个有$n$个顶点的图，有独立数$α$，且对于某个$t\geq5$没有$K_{t+1}$ -子图.证明了$(2α-1)(2t-5)\geq2n-5$ .
Bipartite Minors
我们引入了一个概念的二部未成年人和证明了一个二部类似的瓦格纳定理：一个二部图是平面的当且仅当它不包含$K_{3,3}$作为一个二部未成年人。同样，我们提供了一个禁止的外平面图和森林的小特征。然后，我们建立了二部$(2,2)$ -Laman图--一类包含所有极大二部可平面图的图族的递归刻画。
In 2009, Brown gave a set of conditions which when satisfied imply that a Feynman integral evaluates to a multiple zeta value. One of these conditions is called reducibility, which loosely says there is an order of integration for the Feynman integral for which Brown's techniques will succeed. Reducibility can be abstracted away from the Feynman integral to just being a condition on two polynomials, the first and second Symanzik polynomials. These polynomials can be defined from graphs, and thus reducibility is a property of graphs. We prove that for a fixed number of external momenta and no m…
For positive integers $t$ and $s$, let $\mathcal{K}_t^{-s}$ denote the family of graphs obtained from the complete graph $K_t$ by removing $s$ edges. A graph $G$ has no $\mathcal{K}_t^{-s}$ minor if it has no $H$ minor for every $H\in \mathcal{K}_t^{-s}$. Motivated by the famous Hadwiger's Conjecture, Jakobsen in 1971 proved that every graph with no $\mathcal{K}_7^{-2}$ minor is $6$-colorable; very recently the present authors proved that every graph with no $\mathcal{K}_8^{-4}$ minor is $7$-colorable. In this paper we continue our work and prove that every graph with no $\mathcal{K}_9^{-6}$ m…
Complete and almost complete minors in double-critical 8-chromatic graphs
一个连通的$k$ -色图$G$称为{ \it双临界}，如果对于$G$的所有边$uv$，图$G - u - v$是$(k-2)$ -色图.一个长期存在的猜想Erdés和Lovász指出，完全图是唯一的双临界图。Kawarabayashi，Pedersen和Toft [ \it{Electron. J. Combin.}，17（1）：Research Paper 87，2010]证明了每个具有$k \leq 7$的双临界$k$ -色图包含一个$K_k$子图.它仍然是未知的，是否一个任意的双临界$8$ -chromatic图包含一个$K_8$子，但在本文中，我们证明了任何双临界$8$ -chromatic包含一个$K_8^-$子;这里$K_8^-$表示完全$8$ -图与一个边缘失踪。此外，我们还观察到任何最小度不同于$10$和$11$的双临界$8$ -色图都包含一个子图$K_8$.
An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph wi…
Markov bases of binary graph models of K_4-minor free graphs
图的马尔可夫宽度是图的不变量，定义为二进制列联表的相应图模型的马尔可夫基元素的最大度。我们证明了一个图的马尔可夫宽度最多为4当且仅当它不包含$K_4$作为一个未成年人，回答了一个问题的Reynoldin和Sullivant。给出了$Ω(n^{2-\varepsilon})$阶数关于$K_n$的马尔可夫宽度的下界.
Cliques in Odd-Minor-Free Graphs
本文讨论：（1）图中某类团的个数的界;（2）列出图中所有团的算法。我们提出了一个简单的算法，列出所有团在$n$ -顶点图在O（n）时间每个团。对于O（1）-退化图，如不包括一个固定的子图，我们描述了一个O（n）时间算法列出所有团。证明了不含固定奇子图的图有$O(n^2)$团（这是紧的），并给出了列出所有团的$O(n^3)$时间算法.
Nowhere-zero 4-flows in graphs excluding a proper minor of the Petersen graph
研究最小度至少为三且围长至少为五的几乎 4 - 连通非平面图含\(P/e\)子图情况，利用此结构结果证明有限无桥不含\((P/e)\)子图的多重图有无处零 4 - 流，扩展并补充相关定理，结合多种定理证明，部分由计算机辅助。
We provide a complete structural characterization of $K_{2,4}$-minor-free graphs. The $3$-connected $K_{2,4}$-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains $K_4$ and, for each $n \ge 5$, $2n-8$ nonisomorphic graphs of order $n$. To describe the $2$-connected $K_{2,4}$-minor-free graphs we use $xy$-outerplanar graphs, graphs embeddable in the plane with a Hamilton $xy$-path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, $xy$-outerplanar graphs …
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. While Hadwiger's conjecture does not hold for list-coloring, the linear weakening is conjectured to be true. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and thus is $O(t\sqrt{\log t})$-list-colorable. Recently, the authors and Song proved that every graph with no $K_t$ minor is $O(t(\log t)^β)$-colorable for every $β> \frac 1 4$. Here, we build on that result to show that every graph with no $K_t$ m…
Unavoidable Parallel Minors of 4-Connected Graphs
平行子图是由图的任意序列的边收缩和平行边删除得到的。本文证明了对任意正整数k，每一个足够高阶的内部4-连通图都含有一个平行子图，它同构于在k度顶点上具有完全图的K_{4,k}的一个变分，在k度顶点上具有完全图的k-划分三重扇，k-辐双轮，k-辐带轴双轮，（2k+1）-梯级Mobius之字形梯，（2k）级锯齿形阶梯，或K_k。我们还发现了1-，2-和3-连通图的不可避免的平行子图。
Minor-minimal planar graphs of even branch-width
设k>0为整数，H为射影平面上的次极小图，使得每一条同伦非平凡闭曲线至少与H相交k次，G为H的平面双覆盖，将G提升到射影平面的万有覆叠空间，即球面.证明了G是分支宽度为2k的次极小图。我们还展示了分支宽度为6的次极小可平面图的例子，这些图不是这样产生的。
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