26 research outputs found
List colorings of -partite -graphs
A -uniform hypergraph (or -graph) is -partite if
can be partitioned into sets such that each edge in
contains precisely one vertex from each . In this note, we consider list
colorings for such hypergraphs. We show that for any if each
vertex is assigned a list of size , then admits a
proper -coloring, provided is sufficiently large. Up to a constant
factor, this matches the bound on the chromatic number of simple -graphs
shown by Frieze and Mubayi, and that on the list chromatic number of triangle
free -graphs shown by Li and Postle. Our results hold in the more general
setting of "color-degree" as has been considered for graphs. Furthermore, we
establish a number of asymmetric statements matching results of Alon, Cambie,
and Kang for bipartite graphs.Comment: 12 page
Helly-type problems
In this paper we present a variety of problems in the interface between combinatorics and geometry around the theorems of Helly, Radon, Carathéodory, and Tverberg. Through these problems we describe the fascinating area of Helly-type theorems and explain some of their main themes and goals
Analytic methods for uniform hypergraphs
This paper develops analityc methods for investigating uniform hypergraphs.
Its starting point is the spectral theory of 2-graphs, in particular, the
largest and the smallest eigenvalues of 2-graphs. On the one hand, this simple
setup is extended to weighted r-graphs, and on the other, the
eigenvalues-numbers are generalized to eigenvalues-functions, which encompass
also other graph parameters like Lagrangians and number of edges. The resulting
theory is new even for 2-graphs, where well-settled topics become challenges
again. The paper covers a multitude of topics, with more than a hundred
concrete statements to underpin an analytic theory for hypergraphs. Essential
among these topics are a Perron-Frobenius type theory and methods for extremal
hypergraph problems. Many open problems are raised and directions for possible
further research are outlined.Comment: 71 pages. Corrected wrong claim in the introductio
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Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics
무지개 집합 문제에서의 위상수학적 조합론
학위논문(박사)--서울대학교 대학원 :자연과학대학 수리과학부,2019. 8. 국웅.를 의 공집합이 아닌 부분 집합들의 모임이라 할 때, 의 무지개 집합이란 공집합이 아니며 와 같은 형태로 주어지는 것으로 다음 조건을 만족하는 것을 말한다. 이고 이면 를 만족하며 각 에 대해 이다. 특히 인 경우, 즉 모든 들이 표현되면, 무지개 집합 를 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 be a finite family of non-empty subsets on the ground set . A rainbow set of is a non-empty set of the form with such that for every and for each . If , namely if all is represented, then the rainbow set is called a full rainbow set of .
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 -free graphs . . . . . . . . . . . . . . . . . 44
4.1.3 Chordal graphs . . . . . . . . . . . . . . . . . . . . . . . 49
4.1.4 -free graphs and -free graphs . . . . . . . . . . . . . 50
4.2 -colourable graphs . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Graphs with bounded degrees . . . . . . . . . . . . . . . . . . . . 55
4.3.1 The case . . . . . . . . . . . . . . . . . . . . . . . 56
4.4 A topological approach . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . 67
Abstract (in Korean) 69
Acknowledgement (in Korean) 70Docto
Tverberg's theorem is 50 Years Old: A survey
This survey presents an overview of the advances around Tverberg's theorem, focusing on the last two decades. We discuss the topological, linear-algebraic, and combinatorial aspects of Tverberg's theorem and its applications. The survey contains several open problems and conjectures. © 2018 American Mathematical Society
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Combinatorics, Probability and Computing
The main theme of this workshop was the use of probabilistic
methods in combinatorics and theoretical computer science. Although
these methods have been around for decades, they are being refined all
the time: they are getting more and more sophisticated and powerful.
Another theme was the study of random combinatorial structures,
either for their own sake, or to tackle extremal questions. The workshop
also emphasized connections between probabilistic combinatorics and
discrete probability