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Results tagged with graph-theory
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user 43954
Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
2
votes
Lower bound for the size of a family of sets
Let $n=|\mathcal F'|$. Below we easily show that $n\ge\sqrt{m}$ and I shall look for more refined arguments to improve this bound.
For each natural $N$ put $[N]=\{1,2,\dots,N\}$. For each $i\in [2]$ p …
- 5,409
2
votes
Visualizing the elements of a finite group and does the Gram matrix determine the finite group?
The expression for $k(g,h)$ is rather complicated and it is hard to recover propertied of $G$ from the properties of $k$. But let us try at least to begin.
We have $k(gx,hx)=k(g,h)$ for each $g,h,x\in …
- 5,409
9
votes
Mark some vectors in $\mathbb{R}^n$ in a way that every orthonormal basis has an odd number ...
Peter Mueller provided a negative
answer for $n=4$. Based on it, we show that the answer is negative for any even $n\ge 4$.
Indeed, suppose for a contradiction that the space $V=\mathbb R^n$ admits a …
- 5,409
3
votes
Connected geometric thickness two
I tried to find a required example, but failed (I share my findings below). Nevertheless, it seems rather strange to me if there is no such example, so I hope that it can be constructed.
A natural ide …
- 5,409
10
votes
Accepted
Desargues ten point configuration $D_{10}$ in LaTeX
This example shows that $s\le 2$ and for this $s$, $c\le 3$.
Note that we are lucky with the latter, because there is a configuration for which every combinatorially equivalent realization has at lea …
- 5,409
3
votes
0
answers
200
views
Combinatorial characterizations of complex weight supports
This question is related to my last question and is originally motivated by recent advances in quantum physics.
I am looking for combinatorial characterizations of some algebraically defined families …
- 5,409
10
votes
0
answers
620
views
A rainbow perfect matching in an edge-colored graph with spanning color classes
This question is a sequel of my last question and is eventually motivated by recent advances in quantum physics. Given an even number $n\ge 6$ and a positive integer $k<n$, Claim from the linked quest …
- 5,409
1
vote
Combinatorial equation system with exponentially many equations in quadratic many variables
My answer concerns Question 2. When I was dealing with the system for $n=4$ I noticed that we can split it as follows (unfortunately, I failed to obtain essential advances from this observation).
Let …
- 5,409
3
votes
Accepted
Relationship between minimum vertex cover and matching width
Your suspicion is correct. The following hypergraph $H$ provides a negative answer to your question. Let $V=\{0,1,\dots, 11\}$. Then $V=V_0\cup V_1\cup V_2$, where $V_0=\{0,1,2,3\}$, $V_1=\{4,5,6,7\}$ …
- 5,409
4
votes
Accepted
Independent sets in complement of Kneser graphs
According to [p. 8], Baranyai's theorem [B] implies that the vertex set of the Kneser graph $K(n,k)$ can be partitioned into $\left\lceil\frac{\binom{n}{k}}{\left\lfloor\frac{n}{k}\right\rfloor}\righ …
- 5,409
3
votes
0
answers
208
views
Clique cover number of a generalized Kneser graph $K(n,4,2)$
Recently I attacked this combinatorial question. The value of $m(n)$ introduced in it equals to a clique cover number
of a generalized Kneser graph $KG_{n,4,1}=K(n,4,2)$ (or the chromatic number of it …
- 5,409
2
votes
Generalizations of Planar Graphs
Wagner-Fáry-Stein theorem states that each (finite simple) planar graph admits a straight-line crossing-free plane drawing. On the other hand, each graph (of at most $\frak c$ vertices) admits a strai …
- 5,409
3
votes
2
answers
1k
views
A structure of the group of automorphisms of an infinite binary tree
My friend asked me to ask his question here. Where he can find (a paper or a book) containing a complete description (with the proof) of a structure of the group of automorphisms of an infinite binary …
- 38.6k