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Results tagged with graph-theory
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user 136218
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
Accepted
Looking for a counterexample to a strengthening of the union-closed sets conjecture
A family of counterexamples was found at math.stackexchange using projective planes as I should have expected.
I have copied the answer here for convenience:
Let $n = q^2+q+1$ and consider a projecti …
- 988
4
votes
1
answer
343
views
Looking for a counterexample to a strengthening of the union-closed sets conjecture
[Now crossposted at math.stackexchange]
Let $\mathcal{F} = \{\{x_1, x_2\} : 1 \le x_1 \lt x_2 \le n \}$, $n \ge 8$, and let $\mathcal{G} = \{G_1, \ldots, G_n\}$ be a partition of $\mathcal{F}$ in $n$ …
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1
vote
2
answers
357
views
Lower bound for the size of a family of sets
Consider a family $\mathcal{G} = \{ A_1,B_1,\ldots,B_m \}$ of $m+1$ non-empty finite distinct sets with the following property:
$$A_1 \cap B_k = \emptyset, 1 \le k \le m$$
Let $\mathcal{F} = \{A_1 \cu …
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4
votes
1
answer
155
views
Remove an edge from the Hasse diagram of a finite lattice
If we remove an edge from the Hasse diagram of a finite lattice, as long as any vertex maintain at least one upward edge and at least one downward edge, do we still always have a lattice from the resu …
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3
votes
2
answers
331
views
Algorithm to evaluate "connectedness" of a binary matrix
I have the following problem: given an $m \times n$ binary matrix $A$ like e.g. the following $9 \times 39$ matrix:
111000011100001101000001001111111111000
111000011100001101000000101111111111000
1111 …
- 988
1
vote
1
answer
73
views
Lower bound for the sum of the number of vertices of some subgraphs of a directed graph
Let $G$ be any simple weakly connected directed graph with vertices $V$, $\vert V \vert = n$. Let $V_1, \ldots, V_m$, $m = \binom{n}{k}$ be all subsets of $V$ of size $k$.
Let $C(V_i)$ be the union of …
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1
vote
1
answer
106
views
Non-isomorphic walk-regular graphs with the same number of closed walks at any length
Are there known examples of couples of non-isomorphic walk-regular graphs with adjacency matrix $A_1$ and $A_2$ and such that $(A_1^k)_{i,i} = (A_2^k)_{i,i}$ for all $k \gt 0$?
- 988
0
votes
1
answer
159
views
Isomorphism of two regular hypergraphs
Consider two undirected $k$-regular hypergraphs on $n$ vertices with (see e.g. OEIS A319190). Are the two hypergraphs isomorphic if an only if the two multisets of the sizes of their respective hypere …
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1
vote
0
answers
63
views
Lower bound for the minimum of the maximum frequency of an element - with restrictions
Consider a family $\mathcal{F}$ of non-empty sets, with
$n=|\mathcal{F}|$ sets, $q=\left|\cup\mathcal{F}\right|$ elements in the universe, and $q\le n/4$.
It is known that of the $\binom{n}{2}$ ways t …
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1
vote
1
answer
80
views
Improving a lower bound for the minimum of the maximum frequency of an element in a family o...
[Originally posted at math.stackexchange without answer]
Consider a family $\mathcal{F}$ of $n=|\mathcal{F}|$ sets, $\emptyset \not\in \mathcal{F}$ and an universe $U(\mathcal{F})$ of $q=|U(\mathcal{F …
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1
vote
1
answer
241
views
Graphs with $n$ vertices and $m$ edges and more probable property
Following to my previous question on the same topic, I would like to have some opinions whether the present refinement have some chances to work or is doomed to fail.
Given the positive integers $n$ a …
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1
vote
2
answers
222
views
Do all graphs with $n$ vertices and $m$ edges have a special property?
Given the positive integers $n$ and $m$, consider the set of graphs $\mathcal{G} = \{G=(V,E): |V|=n \land |E|=m\}$.
For which values of $n$ and $m$ does the following requirement hold:
$\forall G \in …
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0
votes
1
answer
97
views
Name and/or class of a graph?
Does the following graph:
(with graph6 string FFzvO) have a name or belong to a special class?
The DIMACS (bliss) format of it is:
p edge 7 14
e 1 4
e 1 5
e 1 6
e 1 7
e 2 4
e 2 5
e 2 6
e 2 7
e 3 4
e …
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1
vote
0
answers
157
views
Given a graph, how to build another "difficult" isomorphic graph from it?
Suppose I want to test an algorithm for graph isomorphism check.
Starting from a given graph, I would do a random permutation of vertices to build the second graph.
Is there a way of making another is …
- 988
1
vote
0
answers
83
views
Literature about graph isomorphism and incidence matrix [closed]
I would like to read some paper, if any, for some classes of graphs, regarding inverting (right/left inverting) the incidence matrix to solve the graph isomorphism problem. Or anyway some known facts …
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