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Results tagged with graph-theory
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user 75
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
Accepted
"Canonical" graph structure on $\text{Hom}(G, H)$
It is trivial that such a maximal $E$ exists, and consists of pairs $\{f,g\}$ such that whenever $x$ and $y$ are adjacent, $f(x)$ and $g(y)$ are adjacent. However, it does not make $\operatorname{Hom …
- 31.2k
7
votes
Hedetniemi's conjecture for graphs with countable chromatic number
No; this is a theorem of Hajnal. Suppose $G$ is a graph such that $\chi(G)$ is infinite, and say that a set of vertices of $G$ is chromatically cofinite if the induced subgraph on the complement is f …
- 31.2k
2
votes
Accepted
Does this version of Hadwiger's conjecture hold for graphs with infinite chromatic number?
First, suppose $G$ has a connected component $C\subseteq G$ with the same chromatic number as $G$. If $C\neq G$, we can take $M=C$. If $G=C$, let $M$ be the subgraph of $G$ obtained by removing all …
- 31.2k
8
votes
Accepted
"Homomorphism fingerprint" for graphs
In any category, if $G$ and $H$ are objects such that there exist monic maps $G\to H$ and $H\to G$, then $|\text{Hom}(X, G)| = |\text{Hom}(X, H)|$ for all $X$. There are plenty of pairs of non-isomor …
- 31.2k
6
votes
Accepted
Infinite graphs isomorphic to their line graph
Note first that $L$ is naturally an endofunctor on the category of graphs and injective graph-homomorphisms that commutes with filtered colimits. Let $G$ be any graph such that there is an embedding …
- 31.2k
39
votes
What is a continuous path?
It sounds to me like what you're looking for is something like Cech (co)homology. The idea is that you can detect what kind of "paths" there are in a space by the combinatorics of which sets in open …
- 31.2k