Ron Aharoni, Eli Berger, Ran Ziv
Konig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalisation of this result, in which each point in one side of the graph is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell.
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The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G)+mu(G) equals its order, then G is a Konig-Egervary graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality alpha(G)=alpha(G^{2}), where G^{2} denotes the second power of G. In particular, we show that a Konig-Egervary graph is square-stable if and only if it has a perfect matching consisting of pendant edges, and in consequence, we deduce that well-covered trees are exactly the square-st…
The stability number of a graph G, is the cardinality of a stable set of maximum size in G. If the stability number of G remains the same upon the addition of any edge, then G is called $α^{+}$-stable. G is a König-Egervary graph if its order equals the sum of its stability number and the cardinality of a maximum matching. In this paper we characterize $α^{+}$-stable König-Egervary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König-Egervary graph is $α^{+}$-stable if and only if either the set of vertices belonging to no maximum stab…
Supermodular Extension of Vizing's Edge-Coloring Theorem
该研究引入相交2/3-超模函数，提出Vizing边着色定理与弱版超模着色定理的统一推广，并借此给出Gupta边着色定理的新证明，丰富了图论与组合优化的着色理论。
Let $\mathsf{TT}^1$ be the combinatorial principle stating that every finite coloring of the infinite full binary tree has a homogeneous isomorphic subtree. Let $\mathsf{RT}^2_2$ and $\mathsf{WKL}_0$ denote respectively the principles of Ramsey's theorem for pairs and weak König's lemma. It is proved that $\mathsf{TT}^1+\mathsf{RT}^2_2+\mathsf{WKL}_0$ is $Π^0_3$-conservative over the base system $\mathsf{RCA}_0$. Thus over $\mathsf{RCA}_0$, $\mathsf{TT}^1$ and Ramsey's theorem for pairs prove the same $Π^0_3$-sentences.
A subset of vertices is a {\it maximum independent set} if no two of the vertices are adjacent and the subset has maximum cardinality. A subset of vertices is called a {\it maximum dissociation set} if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. Zito [J. Graph Theory {\bf 15} (1991) 207--221] proved that the maximum number of maximum independent sets of a tree of order $n$ is $2^{\frac{n-3}{2}}$ if $n$ is odd, and $2^{\frac{n-2}{2}}+1$ if $n$ is even and also characterized all extremal trees with the most maximum independent sets, which solved a …
Let $α(G)$ denote the cardinality of a maximum independent set, while $μ(G)$ be the size of a maximum matching in the graph $G=\left(V,E\right) $. If $α(G)+μ(G)=\left\vert V\right\vert $, then $G$ is a König-Egerváry graph. If $d_{1}\leq d_{2}\leq\cdots\leq d_{n}$ is the degree sequence of $G$, then the annihilation number $h\left(G\right) $ of $G$ is the largest integer $k$ such that $\sum\limits_{i=1}^{k}d_{i}\leq\left\vert E\right\vert $ (Pepper 2004, Pepper 2009). A set $A\subseteq V$ satisfying $\sum \limits_{a\in A} deg(a)\leq\left\vert E\right\vert $ is an annihilation set, if, in addit…
Loebl, Komlós, and Sós conjectured that any graph with at least half of its vertices of degree at least k contains every tree with at most k edges. We propose a version of this conjecture for skewed trees, i.e., we consider the class of trees with at most k edges such that the sizes of the colour classes of the trees have a given ratio. We show that our conjecture is asymptotically correct for dense graphs. The proof relies on the regularity method. Our result implies bounds on Ramsey number of several trees of given skew.
We prove that for any pair of constants $ε>0$ and $Δ$ and for $n$ sufficiently large, every family of trees of orders at most $n$, maximum degrees at most $Δ$, and with at most $\binom{n}{2}$ edges in total packs into $K_{(1+ε)n}$. This implies asymptotic versions of the Tree Packing Conjecture of Gyarfas from 1976 and a tree packing conjecture of Ringel from 1963 for trees with bounded maximum degree. A novel random tree embedding process combined with the nibble method forms the core of the proof.
The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals alpha(G) + mu(G), where mu(G) is the cardinality of a maximum matching in G. In this paper we characterize $α^{++}$-stable graphs, namely, the graphs whose stability numbers are invariant to adding any two edges from their complements. We show that a König-Egerváry graph is $α^{++}$-stable if and only if it has a perfect matching consisting of pendant edges and no …
Sumner's universal tournament conjecture states that any tournament on $2n-2$ vertices contains a copy of any directed tree on $n$ vertices. We prove an asymptotic version of this conjecture, namely that any tournament on $(2+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices. In addition, we prove an asymptotically best possible result for trees of bounded degree, namely that for any fixed $Δ$, any tournament on $(1+o(1))n$ vertices contains a copy of any directed tree on $n$ vertices with maximum degree at most $Δ$.
We conjecture that every $n$-vertex graph of minimum degree at least $\frac k2$ and maximum degree at least $2k$ contains all trees with $k$ edges as subgraphs. We prove an approximate version of this conjecture for trees of bounded degree and dense host graphs. Our work also has implications on the Erd\H os--Sós conjecture and the $\frac 23$-conjecture. We prove an approximate version of both conjectures for bounded degree trees and dense host graphs.
The Strong Nine Dragon Tree Conjecture asserts that for any integers $k$ and $d$ any graph with fractional arboricity at most $k + \frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that for at least one of the forests, every connected component contains at most $d$ edges. We prove this conjecture when $d \leq k+1$. We also prove an approximate version of this conjecture, that is, we prove that for any positive integers $k$ and $d$, any graph with fractional arboricity at most $k + \frac{d}{d+k+1}$ decomposes into $k+1$ forests, such that one for at least one of the forests, every connected …
This paper is a continuation of arXiv:1612.03873. We prove a three-parameter family of identities (Theorem 1.1) involving a version of the Tutte polynomial for directed graphs introduced by Awan and Bernardi in arXiv:1610.01839. A particular case of this family (Corollary 1.6) is the higher-degree generalization of the matrix-tree theorem proved in arXiv:1612.03873, which thus receives a new proof, shorter (and less direct) than the original one. The theory has a parallel version for undirected graphs (Theorem 1.2).
At the core of the Robertson-Seymour theory of graph minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor. Robertson and Seymour prove several versions of this theorem, each stressing some particular aspects needed at a corresponding stage of the proof of the main result of their theory, the graph minor theorem. We prove a new version of this structure theorem: one that seeks to combine maximum applicability with a minimum of technical ado, and which might serve as a canonical version for fu…
A coarse block-cutvertex tree-decomposition
研究连通图块-割点树分解，通过特定方法得到其粗化版本。
A degree version of the Burr-Erdős conjecture on trees
本文证明了 Schelp 提出的 Burr-Erdős 猜想的度版本：对于任意 ε>0 和 Δ≥2，若图 G 有 N≥(2+ε)n 个顶点且最小度 δ(G)≥⌊3N/4⌋，则 G 的每条边任意红蓝染色都包含一个单色的 n 顶点树，且该树的最大度不超过 Δ。实际上，该结论在去掉 εn 项后仍然成立。
We present a new definition of non-ambiguous trees (NATs) as labelled binary trees. We thus get a differential equation whose solution can be described combinatorially. This yields a new formula for the number of NATs. We also obtain q-versions of our formula. We finally generalise NATs to higher dimension.
Loebl, Komlos, and Sos conjectured that if at least half of the vertices of a graph G have degree at least some natural number k, then every tree with at most k edges is a subgraph of G. Our main result is an approximate version of this conjecture for large enough n=|V(G)|, assumed that n=O(k). Our result implies an asymptotic bound for the Ramsey number of trees. We prove that r(T_k,T_m)\leq k+m+o(k+m),as k+m tends to infinity.
Sprouts is a two-player topological game, invented in 1967 by Michael Paterson and John Conway. The game starts with p spots, lasts at most 3p-1 moves, and the player who makes the last move wins. In the misere version of Sprouts, on the contrary, the player who makes the last move loses. Sprouts is a very intricate game, and the first computer analysis in 1991 reached only p=11. New results were made possible in 2007 up to p=32 by using combinatorial game theory: when a position is a sum of independant games, it is possible to replace some of these games by a natural number, called the nimber…
A depth-first search version of Dhar's burning algorithm is used to give a bijection between the parking functions of a graph and labeled spanning trees, relating the degree of the parking function with the number of inversions of the spanning tree. Specializing to the complete graph answers a problem posed by R. Stanley.
The asymptotic version of the Erdős-Sós conjecture and beyond
研究证明了稠密宿主图中关于度数的树包含猜想的渐进行为，并推导出无度数限制的埃尔德什-索斯猜想及稀疏宿主图中有限度数树的结论。
Towards a Dual Version of Woodall's Conjecture for Partial 3-Trees
本文验证了平面有向图中Woodall猜想对偶版本对3-树及围长3的部分3-树成立，并给出3-树反馈弧集的紧上界及k-树的上界。
We formalize Hall's Marriage Theorem in the Lean theorem prover for inclusion in mathlib, which is a community-driven effort to build a unified mathematics library for Lean. One goal of the mathlib project is to contain all of the topics of a complete undergraduate mathematics education. We provide three presentations of the main theorem statement: in terms of indexed families of finite sets, of relations on types, and of matchings in bipartite graphs. We also formalize a version of Kőnig's lemma (in terms of inverse limits) to boost the theorem to the case of countably infinite index sets. We…
A $k$-plane tree is a plane tree whose vertices are assigned labels between $1$ and $k$ in such a way that the sum of the labels along any edge is no greater than $k+1$. These trees are known to be related to $(k+1)$-ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for $k$-noncrossing trees, a similarly defined family of labelled noncrossing trees that …
Halin's Infinite Ray Theorems: Complexity and Reverse Mathematics: Version E
该研究分析Halin无限射线定理等的可计算与证明论复杂度，证明其属于超算术分析定理，提供多个纯数学例子，解答了反推数学领域的公开问题。
Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour (JCTB 2016). In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger's Conjecture for tree-chromatic number. As evidence that our conjecture may be more tractable than Hadwiger's Conjecture, we give a short proof that ev…
Let $\CRF_S$ denote the category of $S$-colored rooted forests, and $\H_{\CRF_S}$ denote its Ringel-Hall algebra as introduced in \cite{KS}. We construct a homomorphism from a $K^+_0 (\CRF_S)$--graded version of the Hopf algebra of noncommutative symmetric functions to $\H_{\CRF_S}$. Dualizing, we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a $K^+_0 (\CRF_S)$--graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao in \cite{Z}.
This paper continues a line of investigation of the Halpern--Läuchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--Läuchli Theorem for one tree of height $κ$ holds whenever $κ$ is strongly inaccessible and the coloring takes less than $κ$ colors. We prove consistency of the Halpern--Läuchli Theorem for finitely many trees of height $κ$, where $κ$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of weak forms of Halpern--\Lauchli\ for trees of height $κ$, whenever $κ$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong…
A half-tree is an edge configuration whose superimposition with a perfect matching is a tree. In this paper, we prove a half-tree theorem for the Pfaffian principal minors of a skew-symmetric matrix whose column sum is zero; introducing an explicit algorithm, we fully characterize half-trees involved. This question naturally arose in the context of statistical mechanics where we aimed at relating perfect matchings and trees on the same graph. As a consequence of the Pfaffian half-tree theorem, we obtain a refined version of the matrix-tree theorem in the case of skew-symmetric matrices, as wel…
We derive tight bounds on the expected weights of several combinatorial optimization problems for random point sets of size $n$ distributed among the leaves of a balanced hierarchically separated tree. We consider {\it monochromatic} and {\it bichromatic} versions of the minimum matching, minimum spanning tree, and traveling salesman problems. We also present tight concentration results for the monochromatic problems.
Furstenberg-Weiss have extended Szemerédi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend the multidimensional Szemerédi theorem by demonstrating the existence of certain arithmetic structures in large subsets of a cartesian product of trees.
Inequalities Connecting the Annihilation and Independence Numbers
本文研究了图的湮灭数与独立数之间的差异，针对树、二部图和König-Egerváry图证明了一系列紧上界不等式。
The VPN Tree Routing Conjecture is a conjecture about the Virtual Private Network Design problem. It states that the symmetric version of the problem always has an optimum solution which has a tree-like structure. In recent work, Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math., 2007) have shown that the conjecture holds when the network is a ring. A shorter proof of the VPN Conjecture for rings was found a few months ago by Grandoni, Kaibel, Oriolo and Skutella (to appear in Oper. Res. Lett., 2008). In their paper, Grandoni et al. introduce another conjecture, called…
Let $Γ$ be a Cayley graph generated by a transposition tree. A natural problem is to understand how the properties of the Cayley graph depend on those of the underlying transposition tree. We focus here on diameter and distance related questions. We examine some related bounds and prove some properties about them. This paper has been withdrawn and a new version with these results and further extensions is at abs/1111.3114
Perfect sorting by reversals, a problem originating in computational genomics, is the process of sorting a signed permutation to either the identity or to the reversed identity permutation, by a sequence of reversals that do not break any common interval. Bérard et al. (2007) make use of strong interval trees to describe an algorithm for sorting signed permutations by reversals. Combinatorial properties of this family of trees are essential to the algorithm analysis. Here, we use the expected value of certain tree parameters to prove that the average run-time of the algorithm is at worst, poly…
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