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pgrepds opened this issue Mar 24, 2023 · 1 comment
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Add Mycielskian construction #238

pgrepds opened this issue Mar 24, 2023 · 1 comment

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@pgrepds
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pgrepds commented Mar 24, 2023

A hereditary family $\mathcal{G}$ of graphs is $\chi$-bounded with $\chi$-binding function $f: \mathbb{N} \to \mathbb{N}$ if $\chi(G) \leq f(\omega(G))$ for all $G$ in $\mathcal{G}$. A classical result of Erdős shows that the gap between the clique number and the chromatic number of a graph can be arbitrarily large. Multiple interesting research questions arise from this result. The most prominent one asks: when is a family of graphs $\chi$-bounded?

The most famous construction that yields a triangle-free graph with an arbitrary large chromatic number is the Mycielskian construction. Obviously, a consequence of this result is that not all graph families are $\chi$-bounded. The Mycielskian contruction is therefore often used as a counterexample in this field of study. Thus, it would make sense to include this graph family as a static construction in Graphs.jl (staticgraphs.jl) and the Grötzsch graph (3rd Mycielskian) (in small graphs.jl).

I'd like to implement both in Graphs.jl, if there is nothing against it.

@aurorarossi
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Hello! There is already an open pull request (#177) for the mycielski operator.

@pgrepds pgrepds closed this as completed Mar 25, 2023
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