Bounding the collapsibility number of simplicial complexes and graphs

We introduce and study a new combinatorial invariant the theta-number $\theta(X)$ of simplicial complexes, and prove that the inequality $\mathcal{C}(X)\leq \theta(X)$ holds for every simplicial complex $X$, where $\mathcal{C}(X)$ denotes the collapsibility number of $X$. We display the advantages of working with the theta-number. Its purely combinatorial formulation enables us to verify the validity of the existing bounds on both Leray and collapsibility numbers as well as provide new bounds involving other parameters. We show that the theta-number, collapsibility and Leray numbers of a vertex decomposable simplicial complex are all equal. Moreover, we prove that the theta-number of the independence complex of a graph $G$ is closely related to its induced matching number $im(G)$ as it happens to the Leray number of such complexes. We identify graph classes where they are equal, and otherwise provide upper bounds involving it. In particular, we prove that the theta-number is bounded from above by $2\sqrt{n\cdot im(G)}$ for every $n$-vertex graph $G$, and in the case of $2K_2$-free graphs, we lower this bound to $2\log n$. Furthermore, we verify that the theta-number is contraction minor monotone on the underlying graph.