Tomas Kaiser, Mickael Montassier, Andre Raspaud
We prove that for any positive integer $k$, the edges of any graph whose fractional arboricity is at most $k + 1/(3k+2)$ can be decomposed into $k$ forests and a matching.
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Spanning Even Linear Forests with Isolated Edges Share the Anti-Ramsey Numbers of Perfect Matchings
本文证明，顶点数$n\boldsymbol{\text{≥}}6$、各分支顶点数为偶数且含孤立边的偶阶线性生成森林，其反拉姆齐数与$n$顶点完美匹配的反拉姆齐数相同。
We consider the problem of covering an input graph $H$ with graphs from a fixed covering class $G$. The classical covering number of $H$ with respect to $G$ is the minimum number of graphs from $G$ needed to cover the edges of $H$ without covering non-edges of $H$. We introduce a unifying notion of three covering parameters with respect to $G$, two of which are novel concepts only considered in special cases before: the local and the folded covering number. Each parameter measures "how far'' $H$ is from $G$ in a different way. Whereas the folded covering number has been investigated thoroughly…
Forests whose matching powers are linear
本文分类了边理想的某个匹配幂具有线性分解性质的赋权有向森林。
A perfect forest is a spanning forest of a connected graph $G$, all of whose components are induced subgraphs of $G$ and such that all vertices have odd degree in the forest. A perfect forest generalised a perfect matching since, in a matching, all components are trees on one edge. Scott first proved the Perfect Forest Theorem, namely, that every connected graph of even order has a perfect forest. Gutin then gave another proof using linear algebra. We give here two very short proofs of the Perfect Forest Theorem which use only elementary notions from graph theory. Both our proofs yield polynom…
We give a simple graph-theoretic proof of a classical result due to C. St. J. A. Nash-Williams on covering graphs by forests. Moreover we derive a slight generalisation of this statement where some edges are preassigned to distinct forests.
A graph is called pseudo-outerplanar if each block has an embedding on the plane in such a way that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, we prove that each pseudo-outerplanar graph admits edge decompositions into a linear forest and an outerplanar graph, or a star forest and an outerplanar graph, or two forests and a matching, or $\max\{Δ(G),4\}$ matchings, or $\max\{\lceilΔ(G)/2\rceil,3\}$ linear forests. These results generalize some ones on outerplanar graphs and $K_{2,3}$-minor-fre…
This work provides a structural characterisation of hereditary graph classes that do not contain a star forest, several graphs obtained from star forests by subset complementation, a union of cliques, and the complement of a union of cliques as induced subgraphs. This provides, for instance, structural results for graph classes not containing a matching and several complements of a matching. In terms of the speed of hereditary graph classes, our results imply that all such classes have at most factorial speed of growth.
We show that the edges of any planar graph of maximum degree at most $9$ can be partitioned into $4$ linear forests and a matching. Combined with known results, this implies that the edges of any planar graph $G$ of odd maximum degree $Δ\ge 9$ can be partitioned into $\tfrac{Δ-1}{2}$ linear forests and one matching. This strengthens well-known results stating that graphs in this class have chromatic index $Δ$ [Vizing, 1965] and linear arboricity at most $\lceil(Δ+1)/2\rceil$ [Wu, 1999].
In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic connectivity in terms of the matching number or edge covering number.
We determine the maximum number of maximal independent sets of arbitrary graphs in terms of their covering numbers and we completely characterize the extremal graphs. As an application, we give a similar result for König-Egerváry graphs in terms of their matching numbers.
Anti-Ramsey Numbers for Spanning Linear Forests of 3-Vertex Paths and Matchings
本文研究了反拉姆齐数AR(n, kP_3 ∪ tP_2)，其中kP_3 ∪ tP_2是k个不相交的3顶点路径和t大小的匹配的线性森林。作者解决了n=3k+2t的情况，对于k≥1，t≥2。
Let G be a bridgeless cubic graph. A well-known conjecture of Berge and Fulkerson can be stated as follows: there exist five perfect matchings of G such that each edge of G is contained in at least one of them. Here, we prove that in each bridgeless cubic graph there exist five perfect matchings covering a portion of the edges at least equal to 215/231 . By a generalization of this result, we decrease the best known upper bound, expressed in terms of the size of the graph, for the number of perfect matchings needed to cover the edge-set of G.
Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs with a 2-factor consisting of two circuits. The first part of this result implies that Berge Conjecture holds for hypohamiltonian cubic graphs.
Let m be a positive integer and let G be a cubic graph of order 2n. We consider the problem of covering the edge-set of G with the minimum number of matchings of size m. This number is called excessive [m]-index of G in literature. The case m=n, that is a covering with perfect matchings, is known to be strictly related to an outstanding conjecture of Berge and Fulkerson. In this paper we study in some details the case m=n-1. We show how this parameter can be large for cubic graphs with low connectivity and we furnish some evidence that each cyclically 4-connected cubic graph of order 2n has ex…
Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lovász local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each…
Matchings and coverings are central topics in graph theory. The close relationship between these two has been key to many fundamental algorithmic and polyhedral results. For mixed graphs, the notion of matching forest was proposed as a common generalization of matchings and branchings. In this paper, we propose the notion of mixed edge cover as a covering counterpart of matching forest, and extend the matching--covering framework to mixed graphs. While algorithmic and polyhedral results extend fairly easily, partition problems are considerably more difficult in the mixed case. We address the p…
A note on matchings and co-matchings in bipartite graphs
研究二分图中匹配与共匹配，对于具有强埃尔德什 - 哈伊纳尔性质的二分图类，在禁止森林及其二分补图的基础上，给出了限制为匹配时\(\varepsilon\)的定量界。
Conflict-free Hypergraph Matchings and Coverings
研究在超图中构造无冲突几乎完美匹配，并扩展为覆盖特定顶点子集的匹配，为广义拉姆齐理论提供简化证明的黑箱工具。
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand their algebraic and combinatorial nature, we first use rooted forests with multiple decoration sets to construct free Hopf algebras with multiple Hochschild 1-cocycle conditions. Applying the universal property of the underlying operated algebras and the method of Gröbner-Shirshov bases, we then construct free objects in the category of matching Rota-Baxter algebras which is a generalization of Rota-Baxter algebras to…
Covering Bridgeless Claw-Free Cubic Graphs with Perfect Matchings
本文证明Berge猜想对无桥无爪三次图成立，并给出四个完美匹配覆盖至少299/315边数的结果。
The complexity of decomposing a graph into a matching and a bounded linear forest
本文研究了图边分解为匹配与有界线性森林或有界星森林的复杂度，证明了前者对所有 $k \ge 3$ 均为 NP 完全，而后者对所有 $k$ 均可多项式求解。
Notes on Equitable Partitions into Matching Forests in Mixed Graphs and into $b$-branchings in Digraphs
本文提出有向图中分支公平划分的两个扩展：混合图中匹配森林的单准则公平划分和数字有向图中 $b$-分支的公平划分，并证明存在性及多面体的整数分解性质。
We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.
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.
We prove that the dimension of the Specht module of a forest $G$ is the same as the normalized volume of the matching polytope of $G$. We also associate to $G$ a symmetric function $s_G$ (analogous to the Schur symmetric function $s_λ$ for a partition $λ$) and investigate its combinatorial and representation-theoretic properties in relation to the Specht module and Schur module of $G$. We then use this to define notions of standard and semistandard tableaux for forests.
Covering the edges of a graph with perfect matchings
本文改进 Lovász 关于 r-图完美匹配边覆盖的结论，证明线性方程组存在整数或 +1/2 系数解，对应线性无关完美匹配，且半数系数数量受 Petersen brick 数约束。
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.
The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan $d$-coverings, Hall, Puder and Sawin introduced the $d$-matching polynomial of a graph $G$, defined as the uniform average of matching polynomials over the set of $d$-sheeted covering graphs of $G$. We prove that a natural multivariate version of the $d$-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the $d$-matching polynomial. Our theorem also includes graphs with loops, thus answerin…
Let $\mathcal{F}$ be a family of $r$-graphs. The Turán number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erdős Matching Conjecture shows that \[ ex_r(n,M_{k+1}^{(r)})= \max\left\{\binom{rk+r-1}{r},\binom{n}{r}-\binom{n-k}{r}\right\}, \] where $M_{k+1}^{(r)}$ represents the $r$-graph consisting of $k+1$ disjoint edges. Motivated by this conjecture, we consider the Turán problem for tight linear forests. A tight linear forest is an $r$-graph whose connected components are all tight paths or isolated verti…
Let $\mathcal{F}$ be a family of graphs. The Turán number $ex(n;\mathcal{F})$ is defined to be the maximum number of edges in a graph of order $n$ that is $\mathcal{F}$-free. In 1959, Erdős and Gallai determined the Turán number of $M_{k+1}$ (a matching of size $k+1$) as follows: \[ ex(n;M_{k+1})= \max\left\{\binom{2k+1}{2},\binom{n}{2}-\binom{n-k}{2}\right\}. \] Since then, there has been a lot of research on Turán number of linear forests. A linear forest is a graph whose connected components are all paths or isolated vertices. Let $\mathcal{L}_{n,k}$ be the family of all linear forests of o…
A linear forest is a forest in which every connected component is a path. The linear arboricity of a graph $G$ is the minimum number of linear forests of $G$ covering all edges. In 1980, Akiyama, Exoo and Harary proposed a conjecture, known as the Linear Arboricity Conjecture (LAC), stating that every $d$-regular graph $G$ has linear arboricity $\lceil \frac{d+1}{2} \rceil$. In 1988, Alon proved that the LAC holds asymptotically. In 1999, the list version of the LAC was raised by An and Wu, which is called the List Linear Arboricity Conjecture. In this article, we prove that the List Linear Ar…
Given a set of paths $P$ we define the \emph{Path Covering with Forest Number} of $P$} (PCFN($P$)) as the minimum size of a set $F$ of forests satisfying that every path in $P$ is contained in at least one forest in $F$. We show that PCFN($P$) is treatable when $P$ is a set of non-crossing shortest paths in a plane graph or subclasses. We prove that if $P$ is a set of non-crossing shortest paths of a planar graph $G$ whose extremal vertices lie on the same face of $G$, then PCFN($P$)\leq 4$, and this bound is tight.
The P-vertex problem for graphs with perfect matchings
本文将二分图的P-顶点问题结果推广到任意图，引入P-顶点覆盖概念，通过极大匹配划分顶点集构造矩阵，借助隐函数定理完成证明。
A path-matching of order $p$ is a vertex disjoint union of nontrivial paths spanning $p$ vertices. Burr and Roberts, and Faudree and Schelp determined the 2-color Ramsey number of path-matchings. In this paper we study the multicolor Ramsey number of path-matchings. Given positive integers $r, p_1, \dots, p_r$, define $R^{PM}(p_1, \dots, p_r)$ to be the smallest integer $n$ such that in any $r$-coloring of the edges of $K_n$ there exists a path-matching of color $i$ and order at least $p_i$ for some $i\in [r]$. Our main result is that for $r\geq 2$ and $p_1\geq \dots\geq p_r\geq 2$, if $p_1\ge…
Extremal problems for star forests and cliques
本文研究同时禁止团 $K_{k+1}$ 与星森林 $(s+1)S_l$ 的 Turán 数，对充分大的 $n$ 确定其确切值并刻画全部极值图。
We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials -- the Chebyshev, Hermite, and Laguerre polynomials -- can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogona…
On covering cubic graphs with three perfect matchings
研究了立方图用三个完美匹配覆盖的效率，证明了对于特定范围内的比值，存在无限多个立方图满足该比值。
A covering code is a subset $\mathcal{C} \subseteq \{0,1\}^n$ with the property that any $z \in \{0,1\}^n$ is close to some $c \in \mathcal{C}$ in Hamming distance. For every $ε,δ>0$, we show a construction of a family of codes with relative covering radius $δ+ ε$ and rate $1 - \mathrm{H}(δ) $ with block length at most $\exp(O((1/ε) \log (1/ε)))$ for every $ε> 0$. This improves upon a folklore construction which only guaranteed codes of block length $\exp(1/ε^2)$. The main idea behind this proof is to find a distribution on codes with relatively small support such that most of these codes have…
Given a graph $T$ and a family of graphs $\mathcal{F}$, the generalized Turán number of $\mathcal{F}$ is the maximum number of copies of $T$ in an $\mathcal{F}$-free graph on $n$ vertices, denoted by $ex(n,T,\mathcal{F})$. When $T = K_r$, $ex(n, K_r, \mathcal{F})$ is a function specifying the maximum possible number of $r$-cliques in an $\mathcal{F}$-free graph on $n$ vertices. A linear forest is a forest whose connected components are all paths and isolated vertices. Let $\mathcal{L}_{k}$ be the family of all linear forests of size $k$ without isolated vertices. In this paper, we obtained the…
Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive integer $k$, we classify all forests $G$ such that $I(G)^{[k]}$ has linear resolution. We also give a combinatorial formula for the regularity of $I(G)^{[2]}$ for any forest $G$.
The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family F (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph K_n with t colors. Another area is to find the minimum number of monochromatic members of F that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for fixed positive integers s,t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of …
A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, \mrbm: Given an edge-colored graph $G$, how large is the largest rainbow matching in $G$? We present several sharp contrasts in the complexity of this problem. We show, among others, that * can be approximated by a polynomial algorithm with approximation ratio $2/3-\eps$. * is APX-complete, even when restricted to properly edge-colored linear forests without a $5$-vertex path, and is solvable in %time $O(m^{3/2})$ on edge-colored $m$-edge polynom…
Nearly optimal packings of equally sized rainbow forests
该论文证明在满足颜色类大小限制的边染色图中，可渐近最优地填充等大小的彩虹森林，解决了相关猜想在给定范围内的情形。
How to see the forest despite the trees
研究离散优化中纳什 - 威廉姆斯和图特定理相关发展，旨在向非专家阐明其内容，也望给专家带来新意，核心方法是阐述相关理论发展，主要贡献是让不同人群了解该领域成果。
Rooted spanning forests in wheel graphs: Fibonacci-Lucas formulas and extremal root configurations
本文研究轮图中的根生成森林，给出基于Fibonacci和Lucas数的计数公式，并解决根配置的极值问题。
Given an arbitrary sequence of non-negative integers $\vecλ=(λ_1,\dots,λ_n)$ and a graph $G$ with vertex set $\{v_1,\dots,v_n\}$, the bounded degree complex, denoted $\text{BD}^{\vecλ}(G)$, is a simplicial complex whose faces are the subsets $H\subseteq E(G)$ such that for each $i \in \{1,\dots,n\}$, the degree of vertex $v_i$ in the induced subgraph $G[H]$ is at most $λ_i$. When $λ_i=k$ for all $i$, the bounded degree complex $\text{BD}^{\vecλ}(G)$ is called the $k$-matching complex, denoted $M_k(G)$. In this article, we determine the homotopy type of bounded degree complexes of forests. In p…
Star-Forest Decompositions of Complete Graphs
本文研究完全几何图的平面星形森林分解问题，通过构造反例推翻了 Pach 等人的猜想，并证明了偶数阶完全抽象图分解为星形森林的具体结构特征。
The notion of $(σ,ρ)$-dominating set generalizes many notions including dominating set, induced matching, perfect codes or independent sets. Bounds on the maximal number of such (maximal, minimal) sets were established for different $σ$ and $ρ$ and different classes of graphs. In particular, Rote showed that the number of minimal dominating sets in trees of order $n$ is at most $95^{\frac{n}{13}}$ and Golovach et Al. computed the asymptotic of the number of $(σ,ρ)$-dominating sets in paths for all $σ$ and $ρ$. Here, we propose a method to compute bounds on the number of $(σ,ρ)$-dominating sets…
On Maximum Induced Forests of the Balanced Bipartite Graphs
本文围绕平衡二部图的最大导出森林问题，研究树笛卡尔积的去圈数与森林数，证明相关猜想、给出下界及等号情形，还推导了树棱柱图与任意图笛卡尔积的去圈数结论。
A conjecture of Berge and Fulkerson (1971) states that every cubic bridgeless graph contains 6 perfect matchings covering each edge precisely twice, which easily implies that every cubic bridgeless graph has three perfect matchings with empty intersection (this weaker statement was conjectured by Fan and Raspaud in 1994). Let $m_t$ be the supremum of all reals $α\le 1$ such that for every cubic bridgeless graph $G$, there exist $t$ perfect matchings of $G$ covering a fraction of at least $α$ of the edges of $G$. It is known that the Berge-Fulkerson conjecture is equivalent to the statement tha…
On decycling and forest numbers of Cartesian products of trees
本文研究了树的笛卡尔积的去环数和森林数，证明了相关不等式并解决了相关猜想。
For some $k \in \mathbb{Z}_{\geq 0}\cup \infty$, we call a linear forest $k$-bounded if each of its components has at most $k$ edges. We will say a $(k,\ell)$-bounded linear forest decomposition of a graph $G$ is a partition of $E(G)$ into the edge sets of two linear forests $F_k,F_\ell$ where $F_k$ is $k$-bounded and $F_\ell$ is $\ell$-bounded. We show that the problem of deciding whether a given graph has such a decomposition is NP-complete if both $k$ and $\ell$ are at least $2$, NP-complete if $k\geq 9$ and $\ell =1$, and is in P for $(k,\ell)=(2,1)$. Before this, the only known NP-complet…
A matching complex of a simple graph $G$ is a simplicial complex with faces given by the matchings of $G$. The topology of matching complexes is mysterious; there are few graphs for which the homotopy type is known. Marietti and Testa showed that matching complexes of forests are contractible or homotopy equivalent to a wedge of spheres. We study two specific families of trees. For caterpillar graphs, we give explicit formulas for the number of spheres in each dimension and for perfect binary trees we find a strict connectivity bound. We also use a tool from discrete Morse theory called the \t…
Ramsey-finiteness for graph pairs: A complete solution to the Burr-Erdős-Faudree-Schelp conjectures
本文证明了两个1981年的猜想，探讨了图对的Ramsey有限性，并反驳了Burr的更强猜想，同时重新表述了Faudree的定理。
We study a competitive optimization version of $α'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted $\Max(G)$ and $\Min(G)$, respectively. We show that always $|\Max(G)-\Min(G)|\le 1$. We obtain a sufficient condition for $\Max(G)=α'(G)$ that is preserved under cartesian product. In general, $\Max(G)\ge \frac23α'(G)$, with equality for many spl…
A maximum stable set in a graph G is a stable set of maximum size. S is a local maximum stable set if it is a maximum stable set of the subgraph of G spanned by the union of S and N(S), where N(S) is the neighborhood of S. A matching M is uniquely restricted if its saturated vertices induce a subgraph which has a unique perfect matching, namely M itself. One theorem of Nemhauser and Trotter Jr., working as a useful sufficient local optimality condition for the weighted maximum stable set problem, ensures that any local maximum stable set of G can be enlarged to a maximum stable set of G. In on…
Refining a classical proof of Whitney, we show that any $4$-connected planar triangulation can be decomposed into a Hamiltonian path and two trees. Therefore, every $4$-connected planar graph decomposes into three forests, one having maximum degree at most $2$. We use this result to show that any Hamiltonian planar triangulation can be decomposed into two trees and one spanning tree of maximum degree at most $3$. These decompositions improve the result of Gonçalves [Covering planar graphs with forests, one having bounded maximum degree. J. Comb. Theory, Ser. B, 100(6):729--739, 2010] that ever…
Resonance Graphs and Perfect Matchings of Graphs on Surfaces
本文研究曲面上图的共振图结构，证明其可作为诱导子图嵌入超立方体，且Clar覆盖多项式等于共振图的立方体多项式，推广了平面图像族的已有结论。
Given two positive integers l and m, with l \le m, an [l,m]-covering of a graph G is a set M of matchings of G whose union is the edge set of G and such that l \le |L| \le m for every matching L of M. An [l,m]-covering M of G is an excessive [l,m]-factorization of G if the cardinality of M is as small as possible. The number of matchings in an excessive [l,m]-factorization of G (or \infty, if G does not admit an excessive [l,m]-factorization) is a graph parameter called the excessive [l,m]-index of G and denoted by χ'[l,m](G). In this paper we study such parameter. Our main result is a general…
Ramanujan Coverings of Graphs
研究有限连通图\(G\)的拉马努金覆盖，通过证明对每个\(r\)存在\(r -\)覆盖且新特征值有上界\(\rho\)，推广了二分拉马努金图的相关结论。核心方法包括复反射群交错多项式族及图匹配多项式推广，丰富了二分拉马努金图族。
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