Alternative polynomial-time algorithm for Bipartite Matching

If $G$ is a bipartite graph, Hall's theorem \cite{H35} gives a condition for the existence of a matching of $G$ covering one side of the bipartition. This theorem admits a well-known algorithmic proof involving the repeated search of augmenting paths. We present here an alternative algorithm, using a game-theoretic formulation of the problem. We also show how to extend this formulation to the setting of balanced hypergraphs

Node-balancing by edge-increments

Suppose you are given a graph $G=(V,E)$ with a weight assignment $w:V\rightarrow\mathbb{Z}$ and that your objective is to modify $w$ using legal steps such that all vertices will have the same weight, where in each legal step you are allowed to choose an edge and increment the weights of its end points by $1$. In this paper we study several variants of this problem for graphs and hypergraphs. On the combinatorial side we show connections with fundamental results from matching theory such as Hall's Theorem and Tutte's Theorem. On the algorithmic side we study the computational complexity of associated decision problems. Our main results are a characterization of the graphs for which any initial assignment can be balanced by edge-increments and a strongly polynomial-time algorithm that computes a balancing sequence of increments if one exists.Comment: 10 page

Connected matchings in special families of graphs.

A connected matching in a graph is a set of disjoint edges such that, for any pair of these edges, there is another edge of the graph incident to both of them. This dissertation investigates two problems related to finding large connected matchings in graphs. The first problem is motivated by a famous and still open conjecture made by Hadwiger stating that every k-chromatic graph contains a minor of the complete graph Kk . If true, Hadwiger\u27s conjecture would imply that every graph G has a minor of the complete graph K n/a(C), where a(G) denotes the independence number of G. For a graph G with a(G) = 2, Thomassé first noted the connection between connected matchings and large complete graph minors: there exists an ? \u3e 0 such that every graph G with a( G) = 2 contains K ?+, as a minor if and only if there exists a positive constant c such that every graph G with a( G) = 2 contains a connected matching of size cn. In Chapter 3 we prove several structural properties of a vertexminimal counterexample to these statements, extending work by Blasiak. We also prove the existence of large connected matchings in graphs with clique size close to the Ramsey bound by proving: for any positive constants band c with c \u3c ¼, there exists a positive integer N such that, if G is a graph with n =: N vertices, 0\u27( G) = 2, and clique size at most bv(n log(n) )then G contains a connected matching of size cn. The second problem concerns computational complexity of finding the size of a maximum connected matching in a graph. This problem has many applications including, when the underlying graph is chordal bipartite, applications to the bipartite margin shop problem. For general graphs, this problem is NP-complete. Cameron has shown the problem is polynomial-time solvable for chordal graphs. Inspired by this and applications to the margin shop problem, in Chapter 4 we focus on the class of chordal bipartite graphs and one of its subclasses, the convex bipartite graphs. We show that a polynomial-time algorithm to find the size of a maximum connected matching in a chordal bipartite graph reduces to finding a polynomial-time algorithm to recognize chordal bipartite graphs that have a perfect connected matching. We also prove that, in chordal bipartite graphs, a connected matching of size k is equivalent to several other statements about the graph and its biadjacency matrix, including for example, the statement that the complement of the latter contains a k x k submatrix that is permutation equivalent to strictly upper triangular matrix

무지개 집합 문제에서의 위상수학적 조합론

학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. 국웅.$\mathcal{F}=\{S_1,\ldots,S_m\}$를 $V$의 공집합이 아닌 부분 집합들의 모임이라 할 때, $\mathcal{F}$의 무지개 집합이란 공집합이 아니며 $S=\{s_{i_1},\ldots,s_{i_k}\} \subset V$와 같은 형태로 주어지는 것으로 다음 조건을 만족하는 것을 말한다. $1\leq i_1<\cdots<i_k \leq m$이고 $j \ne j$이면 $s_{i_j} \ne s_{i_j'}$를 만족하며 각 $j \in [m]$에 대해 $s_{i_j} \in S_{i_j}$이다. 특히 $k=m$인 경우, 즉 모든 $S_i$들이 표현되면, 무지개 집합 $S$를 $\mathcal{F}$의 완전 무지개 집합이라고 한다. 주어진 집합계가 특정 조건을 만족하는 무지개 집합을 가지기 위한 충분 조건을 찾는 문제는 홀의 결혼 정리에서 시작되어 최근까지도 조합수학에서 가장 대표적 문제 중 하나로 여겨져왔다. 이러한 방향으로의 문제를 무지개 집합 문제라고 부른다. 본 학위논문에서는 무지개 집합 문제와 관련하여 위상수학적 홀의 정리와 위상수학적 다색 헬리 정리를 소개하고, (하이퍼)그래프에서의 무지개 덮개와 무지개 독립 집합에 관한 결과들을 다루고자 한다.Let $\mathcal{F}=\{S_1,\ldots,S_m\}$ be a finite family of non-empty subsets on the ground set $V$. A rainbow set of $\mathcal{F}$ is a non-empty set of the form $S=\{s_{i_1},\ldots,s_{i_k}\} \subset V$ with $1 \leq i_1 < \cdots < i_k \leq m$ such that $s_{i_j} \neq s_{i_{j'}}$ for every $j \neq j'$ and $s_{i_j} \in S_{i_j}$ for each $j \in [k]$. If $k = m$, namely if all $S_i$ is represented, then the rainbow set $S$ is called a full rainbow set of $\mathcal{F}$. Originated from the celebrated Hall's marriage theorem, it has been one of the most fundamental questions in combinatorics and discrete mathematics to find sufficient conditions on set-systems to guarantee the existence of certain rainbow sets. We call problems in this direction the rainbow set problems. In this dissertation, we give an overview on two topological tools on rainbow set problems, Aharoni and Haxell's topological Hall theorem and Kalai and Meshulam's topological colorful Helly theorem, and present some results on and rainbow independent sets and rainbow covers in (hyper)graphs.Abstract i 1 Introduction 1 1.1 Topological Hall theorem . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Topological colorful Helly theorem . . . . . . . . . . . . . . . . . 3 1.2.1 Collapsibility and Lerayness of simplicial complexes . . . 4 1.2.2 Nerve theorem and topological Helly theorem . . . . . . . 5 1.2.3 Topological colorful Helly theorem . . . . . . . . . . . . 6 1.3 Domination numbers and non-cover complexes of hypergraphs . . 7 1.3.1 Domination numbers of hypergraphs . . . . . . . . . . . . 10 1.3.2 Non-cover complexes of hypergraphs . . . . . . . . . . . . 10 1.4 Rainbow independent sets in graphs . . . . . . . . . . . . . . . . 12 1.5 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Collapsibility of non-cover complexes of graphs 16 2.1 The minimal exclusion sequences . . . . . . . . . . . . . . . . . . 16 2.2 Independent domination numbers and collapsibility numbers of non-cover complexes of graphs . . . . . . . . . . . . . . . . . . . 21 3 Domination numbers and non-cover complexes of hypergraphs 24 3.1 Proof of Theorem 1.3.4 . . . . . . . . . . . . . . . . . . . . . . . 25 3.1.1 Edge-annihilation . . . . . . . . . . . . . . . . . . . . . . 25 3.1.2 Non-cover complexes for hypergraphs . . . . . . . . . . . 27 3.2 Lerayness of non-cover complexes . . . . . . . . . . . . . . . . . 30 3.2.1 Total domination numbers . . . . . . . . . . . . . . . . . 30 3.2.2 Independent domination numbers . . . . . . . . . . . . . 33 3.2.3 Edgewise-domination numbers . . . . . . . . . . . . . . . 34 3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3.1 Independent domination numbers of hypergraphs . . . . . 35 3.3.2 Independence complexes of hypergraphs . . . . . . . . . . 36 3.3.3 General position complexes . . . . . . . . . . . . . . . . . 37 3.3.4 Rainbow covers of hypergraphs . . . . . . . . . . . . . . 39 3.3.5 Collapsibility of non-cover complexes of hypergraphs . . . 40 4 Rainbow independent sets 42 4.1 Graphs avoiding certain induced subgraphs . . . . . . . . . . . . 42 4.1.1 Claw-free graphs . . . . . . . . . . . . . . . . . . . . . . 42 4.1.2 $\{C_4,C_5, . . . ,C_s\}$-free graphs . . . . . . . . . . . . . . . . . 44 4.1.3 Chordal graphs . . . . . . . . . . . . . . . . . . . . . . . 49 4.1.4 $K_r$-free graphs and $K^{−}_r$-free graphs . . . . . . . . . . . . . 50 4.2 $k$-colourable graphs . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Graphs with bounded degrees . . . . . . . . . . . . . . . . . . . . 55 4.3.1 The case $m < n$ . . . . . . . . . . . . . . . . . . . . . . . 56 4.4 A topological approach . . . . . . . . . . . . . . . . . . . . . . . 64 4.5 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . 67 Abstract (in Korean) 69 Acknowledgement (in Korean) 70Docto

Graphs of low average degree without independent transversals

An independent transversal of a graph G with a vertex partition P is an independent set of G intersecting each block of P in a single vertex. Wanless and Wood proved that if each block of P has size at least t and the average degree of vertices in each block is at most t/4, then an independent transversal of P exists. We present a construction showing that this result is optimal: for any ε>0 and sufficiently large t, there is a family of forests with vertex partitions whose block size is at least t, average degree of vertices in each block is at most (1/4+ε)t, and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld-Wanless-Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version

Realizing degree sequences in parallel

A sequence $d$ of integers is a degree sequence if there exists a (simple) graph $G$ such that the components of $d$ are equal to the degrees of the vertices of $G$. The graph $G$ is said to be a realization of $d$. We provide an efficient parallel algorithm to realize $d$. Before our result, it was not known if the problem of realizing $d$ is in $NC$

Extremal problems on counting combinatorial structures

The fast developing field of extremal combinatorics provides a diverse spectrum of powerful tools with many applications to economics, computer science, and optimization theory. In this thesis, we focus on counting and coloring problems in this field. The complete balanced bipartite graph on $n$ vertices has \floor{n^2/4} edges. Since all of its subgraphs are triangle-free, the number of (labeled) triangle-free graphs on $n$ vertices is at least 2^{\floor{n^2/4}}. This was shown to be the correct order of magnitude in a celebrated paper Erd\H{o}s, Kleitman, and Rothschild from 1976, where the authors furthermore proved that almost all triangle-free graphs are bipartite. In Chapters 2 and 3 we study analogous problems for triangle-free graphs that are maximal with respect to inclusion. In Chapter 2, we solve the following problem of Paul Erd\H{o}s: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. We show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the previously known lower bound. Our proof uses among other tools the Ruzsa-Szemer\'{e}di Triangle Removal Lemma and recent results on characterizing of the structure of independent sets in hypergraphs. This is a joint work with J\'{o}zsef Balogh. In Chapter 3, we investigate the structure of maximal triangle-free graphs. We prove that almost all maximal triangle-free graphs admit a vertex partition $(X, Y)$ such that $G[X]$ is a perfect matching and $Y$ is an independent set. Our proof uses the Ruzsa-Szemer\'{e}di Removal Lemma, the Erd\H{o}s-Simonovits stability theorem, and recent results of Balogh-Morris-Samotij and Saxton-Thomason on the characterization of the structure of independent sets in hypergraphs. The proof also relies on a new bound on the number of maximal independent sets in triangle-free graphs with many vertex-disjoint $P_3$'s, which is of independent interest. This is a joint work with J\'{o}zsef Balogh, Hong Liu, and Maryam Sharifzadeh. In Chapte 4, we seek families in posets with the smallest number of comparable pairs. Given a poset $P$, a family \F\subseteq P is \emph{centered} if it is obtained by `taking sets as close to the middle layer as possible'. A poset $P$ is said to have the \emph{centeredness property} if for any $M$, among all families of size $M$ in $P$, centered families contain the minimum number of comparable pairs. Kleitman showed that the Boolean lattice $\{0,1\}^n$ has the centeredness property. It was conjectured by Noel, Scott, and Sudakov, and by Balogh and Wagner, that the poset $\{0,1,\ldots,k\}^n$ also has the centeredness property, provided $n$ is sufficiently large compared to $k$. We show that this conjecture is false for all $k\geq 2$ and investigate the range of $M$ for which it holds. Further, we improve a result of Noel, Scott, and Sudakov by showing that the poset of subspaces of $\mathbb{F}_q^n$ has the centeredness property. Several open problems are also given. This is a joint result with J\'{o}zsef Balogh and Adam Zsolt Wagner. In Chapter 5, we consider a graph coloring problem. Kim and Park have found an infinite family of graphs whose squares are not chromatic-choosable. Xuding Zhu asked whether there is some $k$ such that all $k$-th power graphs are chromatic-choosable. We answer this question in the negative: we show that there is a positive constant $c$ such that for any $k$ there is a family of graphs $G$ with $\chi(G^k)$ unbounded and $\chi_{\ell}(G^k)\geq c \chi(G^k) \log \chi(G^k)$. We also provide an upper bound, $\chi_{\ell}(G^k)1$. This is a joint work with Nicholas Kosar, Benjamin Reiniger, and Elyse Yeager