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Results tagged with graph-theory
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user 108556
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.
4
votes
Accepted
"Hamiltonicity" of a graph
$H(G)\leq \lvert V(G)\rvert-1$ for every finite connected graph.
Proof. Consider any spanning tree $T$ of $G$, double every edge of $T$, choose any Euler tour $W$ of this auxiliary graph, consider t …
- 6,051
2
votes
Density of Cliques in Random Graphs
If by "density" you do not strictly restrict your question to quantitative results (in the sense of results about the numerical value of the clique density), but rather also more qualitative/structura …
- 6,051
2
votes
Bounding a graph invariant
The supremum $q$ of the quantity $q(G)$ you are interested in, over the class of all finite graphs, is at least $\frac13$.
For the time being, I do neither know whether $q$ is larger than $\frac13$, …
- 6,051
4
votes
Accepted
Complete minors and minimal degree
Not an answer but an observation: for any fixed order $r$ of the complete minor, your question can be answered by a finite search over the set of all graphs of order at most $c_{\mathrm{Kostochka}}\sq …
- 6,051
2
votes
Counting labeled triangle-free graphs on $n$ vertices
user36212 has already essentially answered the question; the state of the art has not yet been pointed out though (by that I mean I miss a mention of the latest relevant publications, and a mention o …
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3
votes
Orthogonal representations of graphs
Thanks to a commenter for pointing out that the following was hasty: the fallacy is of course in the statement that the edgeless graph had no faithful orthogonal representation. I'll leave it at that, …
- 6,051
3
votes
Number of spanning trees: bounds from structural parameters
Kostochka's Upper Bound. It is now almost seven years that no one has mentioned the following widely-known, proved, completely general upper bound in terms of the degree-sequence:
Theorem (A. Kos …
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1
vote
cell complexes and higher graph theory
As to your first question, the term "unfair" is not part of mathematical discourse. Of course one can impose axioms, in principle; what good comes of it, depends on the situation. And of course in the …
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5
votes
Neighborhood fingerprint of a graph
Answer. Arguably the simplest infinite set of counterexamples to "Does the converse hold?" are provided by the Möbius ladder's vis-à-vis the prism graphs. Any pair of such graphs (of equal orders) bea …
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7
votes
Non-isomorphic graphs with bijective graph homomorphisms in both directions between them
Here are two partial answers:
EDIT: (the following is tentative; in light of Jeremy Rickards example, which is vertex-3-connected, and with which I cannot find anything wrong, something must be wron …
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16
votes
Does there exist a graph with maximum degree 8, chromatic number 8, clique number 6?
Relevant footnotes to Fedor Petrov's nice, helpful, and completely correct answer.
Fedor's answer seems essentially unimprovable both in brevity and completeness (it's all there).
I hadn't expected …
- 6,051
4
votes
Accepted
"Eccentricity" in the Definition of Graph Center
Re: 'is it customary'. Yes, this is the customary term, and I don't know any reasonable alternative to using the eccentricity function. The term (with exactly this definition) already occurs on page 3 …
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3
votes
Does anyone know a specific polynomial-time algorithm to detect if a given signed graph cont...
As a consequence of
[Seymour, P.D., The matroids with the max-flow min-cut property, JCT B, 23 (1977), p. 189-222],
a finite signed graph has an odd $K^4$ as signed minor
if and only if
the sy …
- 6,051
5
votes
Applications of Perfect Matching
like to learn more about its applications in other domains - specifically in the "real world" (by that I mean something relatively widely useful instead of confined to a very strict vertical like a …
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4
votes
Accepted
Can we define an isomorphism invariant to measure "dimension" of an undirected simple graph?
Five answers, 'by vague association' and 'lateral thinking' (which is unavoidable for this vague question, I think).
All in all, I think that any definition you will give will have an 'air' of arbitra …
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