On a complete characterization of path-free complexes associated with complete multipartite graphs

Let $G$ be a graph and let ${\text{PF}}_t(G)$ denote the simplicial complex whose faces are vertex subsets whose induced subgraphs contain no path on $t$ vertices. These complexes encode a forbidden-subgraph condition as a family of allowed vertex subsets. In this paper, we study $t$-path-free complexes of complete multipartite graphs. Let $$G=K_{n_1,\dots ,n_m},n_1\le \cdots \le n_m\mathrm{.}$$ We first obtain an explicit structural decomposition of ${\text{PF}}_t(G)$ as a union of join complexes, together with an additional lower-dimensional size-truncation term. Using this decomposition, we show that for $t\le 2n_{m-1}-2$ the complex ${\text{PF}}_t(G)$ is not sequentially Cohen-Macaulay, while for $t\ge 2n_{m-1}-1$ it is vertex decomposable. Consequently, we obtain a complete characterization for complete multipartite graphs: ${\text{PF}}_t(G)$ is vertex decomposable if and only if $t\ge 2n_{m-1}-1$. Equivalently, this is also exactly the range in which ${\text{PF}}_t(G)$ is shellable and sequentially Cohen-Macaulay. We further analyze the topology via a Mayer-Vietoris spectral sequence: for complete bipartite graphs, we determine the full homotopy type as an explicit wedge of spheres in all cases.