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 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 $(\alpha,\beta)$ such that the following holds for every $t>0$: whenever $G$ is a graph with maximum degree $\Delta(G)\leq\alpha t$, and $P$ is a $t$-thick vertex partition of $G$ such that $b(G,P)\leq \beta t$, there exists an IT of $G$ with respect to $P$. Our proof makes use of another previously known criterion for the existence of IT's that involves 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 $b(G,P)\leq t/4$ and the old and frequently applied theorem that if $\Delta(G)\leq t/2$ then an IT exists. Using the same approach, we also extend a theorem of Aharoni, Holzman, Howard and Spr\"ussel, by giving a stability version of the latter result.