Highest scored 'combinatorial-designs+co.combinatorics+graph-theory' questions

55 votes

21 answers

14k views

Linear algebra proofs in combinatorics?

Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...

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Tony Huynh

23 votes

2 answers

3k views

Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?

The following problem is homework of a sort -- but homework I can't do! The following problem is in Problem 1.F in Van Lint and Wilson: Let

$G$ be a graph where every vertex has degree

$d$ . ...

David E Speyer

150k

asked Oct 28, 2010 at 1:42

11 votes

5 answers

489 views

What are efficient pooling designs for RT-PCR tests?

I realize this is long, but hopefully I think it may be worth the reading for people interested in combinatorics and it might prove important to Covid-19 testing. Slightly reduced in edit. The ...

Benoît Kloeckner

14.1k

asked Nov 15, 2020 at 11:11

4 votes

2 answers

216 views

Is the domination number of a combinatorial design determined by the design parameters?

Let

$D$ be a

$(v,k,\lambda)$ -design. By the domination number of

$D$ I mean the domination number

$\gamma(L(D))$ of the bipartite incidence graph of

$D$ . Is

$\gamma(L(D))$ determined only by

$v,k$ , ...

Felix Goldberg

6,832

asked Mar 7, 2014 at 0:25

4 votes

3 answers

663 views

Does an $(x, bx)$ -biregular graph always contain a $x$ -regular bipartite subgraph?

I guess a discrete-mathematics-related question is still welcome in MO since I was new to the community and learned from this amazing past post. The following claim is a simplified and abstract form ...

Yungchen Jen

43

asked Mar 22, 2021 at 16:07

4 votes

1 answer

69 views

Balancing out edge multiplicites in a graph

Let

$G$ be a multigraph with maximum edge multiplicity

$t$ and minimum edge multiplicity

$1$ (so that there is at least one 'ordinary' edge). Is there some simple graph

$H$ such that the

$t$ -fold ...

Peter Dukes

1,071

asked Mar 14, 2014 at 22:37

1 vote

0 answers

80 views

Bounds for smallest non-trivial designs

Given

$s>t\ge 2$ , let

$N(s,t)$ be the smallest integer

$n>s$ such that there exists an “

$(n;s;t;1)$ -design” (i.e., a collection of

$s$ -subsets

$e_1,\dots,e_m$ of

$[n]:=\{1,\dots,n\}$ , such that ...

Zach Hunter

2,874

asked Mar 2 at 11:02

0 votes

0 answers

90 views

Steiner-like systems with large edges and many intersections

Let

$l\geq 3$ be an integer. Is there

$n\in\mathbb{N}$ and a hypergraph

$H=(\{1,\ldots,n\},E)$ with the following properties? for all

$e\in E$ we have

$|e| \geq l$ $e_1\neq e_2 \in E \implies |e_1 \...

Dominic van der Zypen

44.5k

asked Jan 21, 2017 at 15:05

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All Questions

Linear algebra proofs in combinatorics?

Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?

What are efficient pooling designs for RT-PCR tests?

Is the domination number of a combinatorial design determined by the design parameters?

Does an $(x, bx)$ -biregular graph always contain a $x$ -regular bipartite subgraph?

Balancing out edge multiplicites in a graph

Bounds for smallest non-trivial designs

Steiner-like systems with large edges and many intersections

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