Julien Baste, Maximilian Fürst, Dieter Rautenbach
A matching $M$ in a graph $G$ is acyclic if the subgraph of $G$ induced by the set of vertices that are incident to an edge in $M$ is a forest. We prove that every graph with $n$ vertices, maximum degree at most $Δ$, and no isolated vertex, has an acyclic matching of size at least $(1-o(1))\frac{6n}{Δ^2},$ and we explain how to find such an acyclic matching in polynomial time.
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The acyclic matching number of a graph $G$ is the largest size of an acyclic matching in $G$, that is, a matching $M$ in $G$ such that the subgraph of $G$ induced by the vertices incident to an edge in $M$ is a forest. We show that the acyclic matching number of a connected subcubic graph $G$ with $m$ edges is at least $m/6$ except for two small exceptions.
Three well-studied types of subgraph-restricted matchings are induced matchings, uniquely restricted matchings, and acyclic matchings. While it is hard to determine the maximum size of a matching of each of these types, whether some given graph has a maximum matching that is induced or has a maximum matching that is uniquely restricted, can both be decided efficiently. In contrast to that we show that deciding whether a given bipartite graph of maximum degree at most four has a maximum matching that is acyclic is NP-complete. Furthermore, we show that maximum weight acyclic matchings can be de…
The weak acyclic matching property in abelian groups
本文在阿贝尔群中研究弱无环匹配性质，该概念与无环匹配相关，后者由重数函数唯一确定，此前已对所有阿贝尔群解决。
Classifying abelian groups through acyclic matchings
本文研究阿贝尔群中的无环匹配性质，在已有 $\mathbb{Z}^n$ 结果基础上，对所有阿贝尔群是否具有该性质给出完整分类。
The notion of acyclic matching property was provided by Losonczy and it was proved that torsion-free groups admit this property. In this paper, we introduce a duality of acyclic matching as a tool for classification of some Abelian groups, moreover, we study matchings for vector spaces and give a connection between matchings in groups and vector spaces. Our tools mix additive number theory, combinatorics and algebra.
We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating $μ$-hyperfinite equivalence relations admit $μ$-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a c…
The question of finding sets of monomials which are removable from a generic homogeneous polynomial through a linear change in its variables was raised by E. K. Wakeford in 1916. This linear algebra question motivated J. Losonczy to define the concept of acyclic matchings in Z n, and later in abelian groups. In this paper, we give a constructive approach to study the acyclic matchings in cyclic groups. We also introduce the notion of weakly matched subsets and investigate its relation with matchings in abelian groups.
Results and questions on matchings in groups and vector subspaces of fields
本文研究阿贝尔群与域扩张场景下的无环匹配问题，给出有限循环群无环匹配存在性结果，完成具线性无环匹配性质的域扩张分类，提出相关开放问题。
Parameterized Results on Acyclic Matchings with Implications for Related Problems
本文证明无环匹配问题在匹配大小为参数时不存在常数因子FPT近似算法，并给出关于三个低于保证值参数的结果，同时证明该问题关于顶点覆盖数加匹配大小无多项式核。
A recursive construction of an acyclic matching on the independence complex of a graph with a simplicial vertex
本文提出一种递归方法，用于构建图的独立复形上的无环匹配，应用离散Morse理论确定了几种图类的同调类型，并为高效同调计算提供预处理框架。
A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In the paper we consider some generalised acyclic $k$-colourings, namely, we require that each colour class induces an acyclic or bounded degree graph. Mainly we focus on graphs with maximum degree 5. We prove that any such graph has an acyclic $5$-colouring such that each colour class induces an acyclic graph with maximum degree at most 4. We prove that the problem of de…
Motivated by the work of Salvetti and Settepanella we introduce certain total orderings of the faces of any shellable regular CW-complex (called `shelling-type orderings') that can be used to explicitly construct maximum acyclic matchings of the poset of cells of the given complex. Building on an application of this method to the classical zonotope shellings we describe a class of maximum acyclic matchings for the Salvetti complex of a linear complexified arrangement. To do this, we introduce and study a new combinatorial stratification of the Salvetti complex. For the obtained acyclic matchin…
Abstract Orientable Incidence Structure and Algorithms for Finite Bounded Acyclic Categories. II. Data Structure and Fundamental Operations
本文构建了有限有界无环范畴的数据结构以编码抽象可定向关联结构，将其表示为带权有向无环多重图，并从几何、范畴和编程角度研究了其基本操作。
Acyclic matchings on Bruhat intervals and applications to total positivity
本文在任意Bruhat区间上基于反射序显式构造无环完全匹配，并借此证明全非负Springer纤维及全非负投影纤维可缩，从而验证Lusztig猜想并简化全非负部分旗簇正则性的证明。
Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--Erdős conjecture, answering a question of Bucić, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $Δ\geq 2$ and any sufficiently large $n$, there exists an acyclic digraph…
Hefner [K. A. S. Hefner, K. F. Jones, S. -R. Kim, R. J. Lundgren and F. S. Roberts: $(i,j)$ competition graphs, Discrete Applied Mathematics, 32, (1991) 241-262] characterized acyclic digraphs each vertex of which has inderee and outdegree at most two and whose competition graphs are interval. They called acyclic digraphs each vertex of which has inderee and outdegree at most two $(2,2)$ digraphs. In this paper, we study the phylogeny graphs of $(2,2)$ digraphs. Especially, we give a sufficient condition and necessary conditions for $(2,2)$ digraphs having chordal phylogeny graphs. Phylogeny g…
A generalization of incidence relations in abstract polytope has been explored, and parameterized surfaces are used as primers. The abstract orientable incidence structure is defined as an algebraic model of incidence relations, in which some algebraic properties in abtract polytope theory are generalized. The geometric interpretation of abstract orientable incidence structure are also discussed. The orientable incidence structure in a semi-regular normal CW complex are briefly investigated.
Translating between the representations of an acyclic convex geometry of bounded degree
本文研究无环凸几何中不可约闭集与蕴涵基的转换问题，证明在度参数有界时该问题易于处理，并提出了基于增量多项式时间的算法。
The dichromatic number of a digraph $G$ is the smallest integer $χ_a(G)$ such that the vertex set of $G$ can be partitioned into $χ_a(G)$ sets, each of which induces an acyclic subdigraph. This is a generalization of the classic chromatic number of graphs. Here, we investigate the dichromatic number of the cartesian, direct, strong and lexicographic products, giving generalizations of some classic results on the chromatic number of products. More specifically, we prove that the following inequalities, known to hold for the chromatic number of graphs, still hold for the dichromatic number of di…
\emph{Bidirected graphs} (a sort of nonstandard graphs introduced by Edmonds and Johnson) provide a natural generalization to the notions of directed and undirected graphs. By a \emph{weakly (node- or edge-) acyclic} bidirected graph we mean such graph having no (node- or edge-) simple cycles. We call a bidirected graph \emph{strongly acyclic} if it has no cycles (even non-simple). Unlike the case of standard graphs, a bidirected graph may be weakly acyclic but still have non-simple cycles. Testing a given bidirected graph for weak acyclicity is a challenging combinatorial problem, which also …
Three-cuts are a charm: acyclicity in 3-connected cubic graphs
针对Mazzuoccolo 2013年提出的三个猜想中更强的第二个猜想（存在两个完美匹配的并的补集为无环子图），证明其虽有2-割反例，但在循环3-边连通立方图中成立。
A matching $M$ in a graph $G$ is $r$-degenerate if the subgraph of $G$ induced by the set of vertices incident with an edge in $M$ is $r$-degenerate. Goddard, Hedetniemi, Hedetniemi, and Laskar (Generalized subgraph-restricted matchings in graphs, Discrete Mathematics 293 (2005) 129-138) introduced the notion of acyclic matchings, which coincide with $1$-degenerate matchings. Solving a problem they posed, we describe an efficient algorithm to determine the maximum size of an $r$-degenerate matching in a given chordal graph. Furthermore, we study the $r$-chromatic index of a graph defined as th…
Competition-common enemy graphs of degree-bounded digraphs
本文完整刻画了⟨2,2⟩有向图的竞争-公共敌图（CCE图），证明分量数≤7的无环⟨2,2⟩有向图的CCE图为区间图且界紧，拓展了度有界有向图相关图类的研究。
In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms. We also present infinite families of prime numbers p such that Z/pZ does not have the acyclic matching property. Finally, we introduce the linear version of acyclic match…
Finite obstruction sets for lower ideals in the minor order are guaranteed to exist by the Graph Minor Theorem. It has been known for several years that, in principle, obstruction sets can be mechanically computed for most natural lower ideals. In this paper, we describe a general-purpose method for finding obstructions by using a bounded treewidth (or pathwidth) search. We illustrate this approach by characterizing certain families of cycle-cover graphs based on the two well-known problems: $k$-{\sc Feedback Vertex Set} and $k$-{\sc Feedback Edge Set}. Our search is based on a number of algor…
On hypergraph Turán problems with bounded matching number
针对匹配数有界的超图场景，本文开展类似Turán问题的研究，得到若干精确结果，拓展了Alon、Frankl与Gerbner在普通图领域的相关研究。
$\mathcal{P}$-matchings Parameterized by Treewidth
本文以树宽为参数，从参数化复杂性角度研究由诱导子图性质 $\mathcal{P}$ 定义的 $\mathcal{P}$-匹配及其计算问题。
A linear $r$-uniform hypergraph is called acycilc if it can be constructed starting from one single edge then at each step adding a new edge that intersect the union of the vertices of the previous edges in at most one vertex. Recently, Gyárfás, Ruszinkó and Sárk\''{o}zy initiated the study of the linear Turán numbers of acyclic linear triple systems. In this paper, we extend their results to linear quadruple systems. Here, we concentrate on small trees, paths and matchings. For the case of small trees, we find that for a linear tree $T$, $ex^{lin}_{4}(n,T)$ relates to difficult problems on St…
We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.
Spectral radius and maximum matching covered graphs with bounded matching number
本文给出了具有有界匹配数的图成为最大匹配覆盖图的尖锐谱半径条件。
Monotone Paths on Acyclic 3-Regular Graphs
本研究改进了3-多面体单调路径数量上界的已有结果，将适用范围拓展至更大的无环3-正则图类，证明其上界为$c \cdot 1.6779^n$（$c$为通用常数），优于此前的$(1+φ)^n$上界。
On Turán problems with bounded matching number
针对匹配数至多为s的n顶点无F图的最大边数这一由Alon和Frankl新近提出的Turán型问题，本文在固定F和s的条件下确定了其相差常数加项的数值，并得到若干精确结论。
For a graph $G$, let $ν_s(G)$ be the induced matching number of $G$. We prove that $ν_s(G) \geq \frac{n(G)}{(\lceil\fracΔ{2}\rceil+1) (\lfloor\fracΔ{2}\rfloor+1)}$ for every graph of sufficiently large maximum degree $Δ$ and without isolated vertices. This bound is sharp. Moreover, there is polynomial-time algorithm which computes induced matchings of size as stated above.
Directed Acyclic Outerplanar Graphs Have Constant Stack Number
该研究提出有向$H$-划分新技术，证明有向无环外平面图具有常数栈数，解决了多个相关猜想与问题，同时构造出栈数无界的有向无环2-树族反驳了另一猜想。
Flow polytopes for extensions of bipartite graphs
本文针对最小度至少为2的二分图，构造其扩展图和几乎度胡须图，证明扩展图的流多胞体的归一化体积等于胡须图中匹配数，并细化得到Ehrhart $h^*$-多项式等于无符号匹配多项式。
The chromatic number of a graph is the minimum $k$ such that the graph has a proper $k$-coloring. There are many coloring parameters in the literature that are proper colorings that also forbid bicolored subgraphs. Some examples are $2$-distance coloring, acyclic coloring, and star coloring, which forbid a bicolored path on three vertices, bicolored cycles, and a bicolored path on four vertices, respectively. This notion was first suggested by Grünbaum in 1973, but no specific name was given. We revive this notion by defining an $H$-avoiding $k$-coloring to be a proper $k$-coloring that forbid…
Excluding a clique or a biclique in graphs of bounded induced matching treewidth
本文研究有界诱导匹配树宽图类的组合性质，证明排除固定双团可推出有界树独立数，排除固定团可推出有界色数，从而证实 Lima 等人提出的两个猜想。
On generalized Tur{á}n problems with bounded matching number and circumference
针对 Alon 和 Frankl 提出的有界匹配数 Turán 问题，本文确定了排除长环族 $C_{\geq k}$ 时 $K_r$ 的广义 Turán 数。
On generalized Turán problems with bounded matching number
本文研究了具有受限匹配数的广义Turán问题，确立了相关的稳定性结果，并利用这些结果给出了当禁用图F为任意非二部图或k顶点路径时广义Turán数的精确值。
We present a family of complete acyclic Morse matchings on the face lattice of a hypersimplex. Since a hypersimplex is a convex polytope, there is a natural way to form a CW complex from its faces. In a future paper we will utilize these matchings to classify every subcomplex whose reduced homology groups are concentrated in a single degree and describe a homology basis for each of them.
Making an oriented graph acyclic using inversions of bounded or prescribed size
通过有限或预设大小的翻转使有向图无环，研究了p-翻转和≤p-翻转的可计算性和复杂性。
On the number and size of Markov equivalence classes of random directed acyclic graphs
本文研究了均匀随机有向无环图的 Markov 等价类，利用塔分解证明了等价类与图的比值趋于正常数，且典型等价类大小有界并具有超多项式尾部。
For positive integers k,n, we investigate the simplicial complex NM_k(n) of all graphs G on vertex set [n] such that every matching in G has size less than k. This complex (along with other associated cell complexes) is found to be homotopy equivalent to a wedge of spheres. The number and dimension of the spheres in the wedge are determined, and (partially conjectural) links to other combinatorially defined complexes are described. In addition we study for positive integers r,s and k the simplicial complex BNM_k(n) of all bipartite graphs G on bipartition [r] \cup [\bar{s}] such that there is …
Extensions of results on phylogeny graphs of degree bounded digraphs
该研究延续贝叶斯网络证据传播相关的系统发育图研究方向，扩展了已有成果，给出弦$(i,2)$系统发育图的必要条件，并通过列出禁用诱导子图给出$(i,j)$系统发育图的必要条件。
We show the existence of rainbow perfect matchings in $μn$-bounded edge colourings of Dirac bipartite graphs, for a sufficiently small $μ>0$. As an application of our results, we obtain several results on the existence of rainbow $k$-factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree.
Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number
研究具有有界匹配数的均匀超图的张量谱稳定性，通过建立定理证明匹配数至多为$\beta$且张量谱半径接近最大值的超图在结构上接近极值超图$S_{n,k,\beta}$，还应用此结果为厄尔多斯匹配猜想谱版本给出新证明。
We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be a diagonal matrix of the degrees of $G$. In 2017, Nikiforov defined the $A_α$-matrix of $G$ as \begin{equation*} A_α(G)=αG)+(1-α)A(G), \end{equation*}d where $α\in[0,1]$ is an arbitrary real number. The largest eigenvalue of $A_α(G)$ is called the $A_α$-spectral radius of $G$. Let $n$, $t$, $k$ be positive integers, satisfying $t\geq1$, $k\geq2$, $n\geq k+2$, and $n\equiv k$ (mod $2$). In this paper, for $α\in[0,\frac{1}{2}]$, we determine the extremal graphs with the maximum $A_α$-spectral radius among all $t$-connected graphs …
On generalized Turán problems with bounded matching number
本文将广义Turán问题中禁止$F$与匹配$M_{s+1}$且$H=K_r$的研究推广到$H$为任意图的情形，扩展了有界匹配数条件下广义Turán数的研究范围。
Ramsey numbers of digraphs with local edge structure
该研究针对有向图的Ramsey数线性性问题，提出分级带宽刻画局部边结构，证明有界度无环且分级带宽有界的有向图定向Ramsey数线性，统一推广了多项已有结果。
A graph is called 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we establish a local property of 1-planar graphs which describes the structure in the neighborhood of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some new classes of light graphs in 1-planar graphs with the bounded degree are found. Therefore, two open problems presented by Fabrici and Madaras [The structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are solved. Furthermore, we prove that each 1-planar graph $G$ with …
Competition graphs of degree bounded digraphs
本文引入放宽无环性条件的 $\langle i,j \rangle$ 有向图概念，研究其竞争图，给出了无环图成为该竞争图的充要条件、弦图的禁用子有向图刻画及竞争图族间的集合包含关系。
Triple systems with bounded matching number: some constructions and exact Turán number
本文推翻了关于避免特定3-图和匹配的图论数的猜想，并确定了某些3-图的精确图论数。
Given a graph $G$ on the vertex set $V$, the non-matching complex of $G$, $\mathsf{NM}_k(G)$, is the family of subgraphs $G' \subset G$ whose matching number $ν(G')$ is strictly less than $k$. As an attempt to generalize the result by Linusson, Shareshian and Welker on the homotopy types of $\mathsf{NM}_k(K_n)$ and $\mathsf{NM}_k(K_{r,s})$ to arbitrary graphs $G$, we show that (i) $\mathsf{NM}_k(G)$ is $(3k-3)$-Leray, and (ii) if $G$ is bipartite, then $\mathsf{NM}_k(G)$ is $(2k-2)$-Leray. This result is obtained by analyzing the homology of the links of non-empty faces of the complex $\mathsf…
Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $ν(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $ν(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $ν(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $ν(G) \le…
For a set of graphs $\mathcal{F}$, let $\ex(n,\mathcal{F})$ and $\spex(n,\mathcal{F})$ denote the maximum number of edges and the maximum spectral radius of an $n$-vertex $\mathcal{F}$-free graph, respectively. Nikiforov ({\em LAA}, 2007) gave the spectral version of the Turán Theorem by showing that $\spex(n, K_{k+1})=λ(T_{k}(n))$, where $T_k(n)$ is the $k$-partite Turán graph on $n$ vertices. In the same year, Feng, Yu and Zhang ({\em LAA}) determined the exact value of $\spex(n, M_{s+1})$, where $M_{s+1}$ is a matching with $s+1$ edges. Recently, Alon and Frankl~(arXiv2210.15076) gave the e…
Bounded degree graphs and hypergraphs with no full rainbow matchings
本文通过构造特定边染色的多超图，证明了全彩虹匹配定理的最优性，并推翻了相关猜想，同时展示了颜色度推广的Galvin定理不成立。
Counting sunflowers in hypergraphs with bounded matching number and Erdős Matching Conjecture in the $(t,k)$-norm
研究在匹配数受限的超图中太阳花的最大计数问题，通过移位方法证明移位操作不减少太阳花数量，并推广了Erdős匹配猜想在$(r-1,k)$-范数中的结果。
We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.
Dichotomies for Maximum Matching Cut: $H$-Freeness, Bounded Diameter, Bounded Radius
研究 Maximum Matching Cut 的计算复杂性，在有界直径、有界半径和无 $H$ 图上给出完整二分性，并据此证明 Disconnected Perfect Matching 在无 $(P_6+sP_2)$ 图上多项式可解。
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