Degree criteria and stability for independent transversals

An 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 (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 $t$-thick, meaning all blocks have size at least $t$. One such result, obtained recently by Wanless and Wood, is based on the maximum average block degree $b\left(G,P\right)=\max \left\{{\sum }_{u\in U}d\left(u\right)∕\mid U\mid :U\in P\right\}$. They proved that if $b\left(G,P\right)\le t∕4$ 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 $\left(\alpha ,\beta \right)$ such that the following holds for every $t>0$: whenever $G$ is a graph with maximum degree $\Delta \left(G\right)\le \alpha t$, and $P$ is a $t$-thick vertex partition of $G$ such that $b\left(G,P\right)\le \beta t$, there exists an IT of $G$ with respect to $P$. Our proof makes use of another previously known criterion for the existence of ITs that involve the topological connectedness of the independence complex of graphs, and establishes a general technical theorem on the structure of graphs for which this parameter is bounded above by a known quantity. Our result interpolates between the criterion $b\left(G,P\right)\le t∕4$ and the old and frequently applied theorem that if $\Delta \left(G\right)\le t∕2$ then an IT exists. Using the same approach, we also extend a theorem of Aharoni, Holzman, Howard and Sprüssel, by giving a stability version of the latter result.