scholarly journals A Paintability Version of the Combinatorial Nullstellensatz, and List Colorings of $k$-partite $k$-uniform Hypergraphs

10.37236/448 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Upper Bound ◽  
Bipartite Graphs ◽  
Combinatorial Nullstellensatz ◽  
List Coloring ◽  
Line List ◽  
Uniform Hypergraphs ◽  
Coloring Number ◽  
Subject Specific ◽  
List Colorings ◽  
On Line

We study the list coloring number of $k$-uniform $k$-partite hypergraphs. Answering a question of Ramamurthi and West, we present a new upper bound which generalizes Alon and Tarsi's bound for bipartite graphs, the case $k=2$. Our results hold even for paintability (on" line list colorability). To prove this additional strengthening, we provide a new subject"=specific version of the Combinatorial Nullstellensatz.

scholarly journals On-line list coloring of matroids

2017 ◽  
Vol 217 ◽  
pp. 353-355 ◽  
Author(s):  
Michał Lasoń ◽  
Wojciech Lubawski
Keyword(s):  
List Coloring ◽  
Line List ◽  
On Line

scholarly journals Deferred On-Line Bipartite Matching

10.37236/5756 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Jakub Kozik ◽  
Grzegorz Matecki
Keyword(s):  
Competitive Ratio ◽  
Bipartite Graphs ◽  
Deterministic Algorithm ◽  
The Other ◽  
Bipartite Matching ◽  
Classical Version ◽  
New Model ◽  
On Line

We present a new model for the problem of on-line matching on bipartite graphs. Suppose that one part of a graph is given, but the vertices of the other part are presented in an on-line fashion. In the classical version, each incoming vertex is either irrevocably matched to a vertex from the other part or stays unmatched forever. In our version, an algorithm is allowed to match the new vertex to a group of elements (possibly empty). Later on, the algorithm can decide to remove some vertices from the group and assign them to another (just presented) vertex, with the restriction that each element belongs to at most one group. We present an optimal (deterministic) algorithm for this problem and prove that its competitive ratio equals $1-\pi/\cosh(\frac{\sqrt{3}}{2}\pi)\approx 0.588$.

scholarly journals On the Strong Equitable Vertex 2-Arboricity of Complete Bipartite Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1778
Author(s):  
Fangyun Tao ◽  
Ting Jin ◽  
Yiyou Tu
Keyword(s):  
Upper Bound ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Equitable Partition ◽  
Special Cases ◽  
Vertex Set ◽  
Complete Bipartite Graphs

An equitable partition of a graph G is a partition of the vertex set of G such that the sizes of any two parts differ by at most one. The strong equitable vertexk-arboricity of G, denoted by vak≡(G), is the smallest integer t such that G can be equitably partitioned into t′ induced forests for every t′≥t, where the maximum degree of each induced forest is at most k. In this paper, we provide a general upper bound for va2≡(Kn,n). Exact values are obtained in some special cases.

scholarly journals Application of polynomial method to on-line list colouring of graphs

2012 ◽  
Vol 33 (5) ◽  
pp. 872-883 ◽  
Author(s):  
Po-Yi Huang ◽  
Tsai-Lien Wong ◽  
Xuding Zhu
Keyword(s):  
Polynomial Method ◽  
Line List ◽  
On Line

scholarly journals Dynamic list coloring of bipartite graphs

2010 ◽  
Vol 158 (17) ◽  
pp. 1963-1965 ◽  
Author(s):  
Louis Esperet
Keyword(s):  
Bipartite Graphs ◽  
List Coloring

scholarly journals Bounding the Number of Common Zeros of Multivariate Polynomials and Their Consecutive Derivatives

2018 ◽  
Vol 28 (2) ◽  
pp. 253-279
Author(s):  
O. GEIL ◽  
U. MARTÍNEZ-PEÑAS
Keyword(s):  
Upper Bound ◽  
Existence And Uniqueness ◽  
Combinatorial Nullstellensatz ◽  
Finite Collection ◽  
Multivariate Polynomials ◽  
Grobner Basis ◽  
Evaluation Codes ◽  
Number Of Zeros ◽  
Interpolating Polynomials ◽  
Common Zeros

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

scholarly journals Triangle-degrees in graphs and tetrahedron coverings in 3-graphs

2020 ◽  
pp. 1-25
Author(s):  
Victor Falgas-Ravry ◽  
Klas Markström ◽  
Yi Zhao
Keyword(s):  
Upper Bound ◽  
Vertex Degree ◽  
Covering Problem ◽  
Uniform Hypergraphs ◽  
Vertex Graph ◽  
Optimal Bounds ◽  
Special Instance ◽  
Tripartite Graphs

Abstract We investigate a covering problem in 3-uniform hypergraphs (3-graphs): Given a 3-graph F, what is c1(n, F), the least integer d such that if G is an n-vertex 3-graph with minimum vertex-degree $\delta_1(G)>d$ then every vertex of G is contained in a copy of F in G? We asymptotically determine c1(n, F) when F is the generalized triangle $K_4^{(3)-}$ , and we give close to optimal bounds in the case where F is the tetrahedron $K_4^{(3)}$ (the complete 3-graph on 4 vertices). This latter problem turns out to be a special instance of the following problem for graphs: Given an n-vertex graph G with $m> n^2/4$ edges, what is the largest t such that some vertex in G must be contained in t triangles? We give upper bound constructions for this problem that we conjecture are asymptotically tight. We prove our conjecture for tripartite graphs, and use flag algebra computations to give some evidence of its truth in the general case.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

scholarly journals On Partitioning and Packing Products with Rectangles

1994 ◽  
Vol 3 (4) ◽  
pp. 429-434 ◽  
Author(s):  
Rudolf Ahlswede ◽  
Ning Cai
Keyword(s):  
Upper Bound ◽  
Complete Graphs ◽  
Minimal Size ◽  
Uniform Hypergraphs ◽  
Image Position ◽  
Special Case

In [1] we introduced and studied for product hypergraphs where ℋi = (i,ℰi), the minimal size π(ℋn) of a partition of into sets that are elements of . The main result was thatif the ℋis are graphs with all loops included. A key step in the proof concerns the special case of complete graphs. Here we show that (1) also holds when the ℋi are complete d-uniform hypergraphs with all loops included, subject to a condition on the sizes of the i. We also present an upper bound on packing numbers.

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