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Results tagged with graph-theory
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user 8008
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.
3
votes
Counting cycle vertex covers on hypercube
The question doesn't ask for exact values in small cases. But it also doesn't NOT ask for them. I wonder about exact values as far as possible, given as nicely as possible, which certainly isn't very …
- 30.1k
5
votes
Accepted
How random are random spanning trees?
This is an interesting question and more subtle than it first appears. One question is the behavior of minimal cost spanning tree for the complete graph. For the question at hand it is worth first con …
- 4,713
1
vote
Odd partition with extra properties
In fact it follows from the proof by @PeterTaylor that there are no solutions over the reals. Indeed, if we drop condition 3 then we can have solutions, $k$ is odd and all the $a_i$ are equal. But tha …
- 30.1k
1
vote
Graphs constructed from sums of perfect matchings
Deciding if a given graph (or type of graph) can be constructed this way would seem the more likely question. When it can, there is sometimes a name for a particular construction of this type. The ci …
- 30.1k
4
votes
Generalization of friendship theorem:n vertices, any m vertices have exactly one neighbor
Assume $ m \lt n$ and call a graph with this property an $(m,n)$ graph. As you already know,
A set of $k$ disjoint edges ( a one factor) is a $(1,2k)$ graph.
$k$ triangles sharing one common vertex …
- 461
10
votes
Why are the numbers counting "ever-closer" lattice paths so round?
In fact the numbers $a_k$ in the various factors $1+a_k$ are multiples of $4$ and typically small ones.
$a_k=0$ unless every prime of the form $4m-1$ dividing $k$ occurs to an even power. If so, $$a_k …
- 30.1k
4
votes
Sum of degree differences for simple graphs
I will guess that the optimum occurs for $k$ isolated vertices and a complete graph on the other $n-k$ where $k=\lfloor\frac{n+1}5\rfloor.$ The same count occurs for $k$ vertices of degree $n-1$ and n …
- 30.1k
1
vote
Identifying the edges that are essential for biconnectivity
Given a bi-connected graph $G$, say an edge $e$ is destructive if $G-e$ has a cut-vertex. Say that $e$ is $(T,F_1,F_2)$-critical if there is a spanning tree $T$ and forests $F_1,F_2$ as in you descri …
- 30.1k
0
votes
A specific collection of subgraphs in $K_{70, 70}$
I like the idea of @dvitek to use pairs of multisets of partitions as a data structure for these $K_{70,70}$ decompositions. Let me repeat the idea since it partly lives in comments.
A $K_{70,70}$ de …
- 30.1k
1
vote
Size of sets in linear intersection structures with large coloring number
I think the answer is yes simply because the only examples are finite projective planes.
Do you know otherwise? Is this what you suspect? If so, why not ask that?
Let me recast things in anothe …
- 30.1k
1
vote
Any results concerning the numbers of vertices and edges to form fixed number of cliques in ...
It might be clearer to say that you have a set of points and $t$ blocks each of size $s$ (I suppose no two totally identical).
What can be said about the range of possible quadruples $(s,t,a,b)$ …
- 30.1k
8
votes
Accepted
What makes Graph invariants so useful/important?
We probably wouldn’t ask what makes graph properties useful. In many ways we consider isomorphic graphs as “the same.” Invariants are just properties that respect this sameness. The specific vertex …
- 30.1k
8
votes
Moore graphs and finite projective geometry
Adding to the allure of this deadly siren song is the fact that there are constructions of this sort for the Moore graph of degree $3$ (the Petersen Graph with 10 vertices and independence number $4$) …
- 30.1k
1
vote
Generalization: (The "number" of) smaller sized clusters in large random binary matrices fol...
I don't see that it has anything to do with Benford's law. That has to do with the frequency of $k$ as the leading digit for lists of a certain sort, like the heights of 100 mountains measured in feet …
- 30.1k
0
votes
Accepted
Why is number of single cell clusters always greatest in a random matrix?
Here is a revised answer that might be clearer:
You define white clusters but I'll just look at black clusters since that is what your data does, and it implies the other interpretation (counts of mo …
- 30.1k