Degree criteria and stability for independent transversals ...
An \emph{independent transversal} (IT) in a graph $G$ with a given vertex partition $P$ is an independent set of vertices of $G$ (i.e. it induces no edges), that consists of one vertex from each part (\emph{block}) of $P$. Over the years, various criteria have been established that guarantee the exi...
| Main Authors: | , |
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| Format: | Report |
| Language: | unknown |
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arXiv
2023
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| Subjects: | |
| Online Access: | https://dx.doi.org/10.48550/arxiv.2305.10595 https://arxiv.org/abs/2305.10595 |
| Summary: | An \emph{independent transversal} (IT) in a graph $G$ with a given vertex partition $P$ is an independent set of vertices of $G$ (i.e. it induces no edges), that consists of one vertex from each part (\emph{block}) of $P$. Over the years, various criteria have been established that guarantee the existence of an IT, often given in terms of $P$ being $t$-\emph{thick}, meaning all blocks have size at least $t$. One such result, obtained recently by Wanless and Wood, is based on the \emph{maximum average block degree} $b(G,P)=\max\{\sum_{u\in U} d(u)/|U| : U \in P\}$. They proved that if $b(G,P)\leq t/4$ then an IT exists. Resolving a problem posed by Groenland, Kaiser, Treffers and Wales (who showed that the ratio $1/4$ is best possible), here we give a full characterization of pairs $(α,β)$ such that the following holds for every $t>0$: whenever $G$ is a graph with maximum degree $Δ(G)\leqαt$, and $P$ is a $t$-thick vertex partition of $G$ such that $b(G,P)\leq βt$, there exists an IT of $G$ with respect to ... |
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