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

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The paper characterizes $t$-path-free complexes $\PF_t(G)$ of complete multipartite graphs $G=K_{n_1,\dots,n_m}$, showing $\PF_t(G)$ is vertex decomposable if and only if $t\ge 2n_{m-1}-1$, which also marks the threshold for sequential Cohen-Macaulayness. For complete bipartite graphs, the authors determine the homotopy type as a wedge of spheres in all cases.

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Exploring path-free complexes in complete multipartite graphs could potentially be useful for understanding and optimizing real-time rendering of complex graph structures.

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arXiv:2607.05358v1 Announce Type: new Abstract: Let $G$ be a graph and let $\PF_t(G)$ 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 \[ G=K_{n_1,\dots,n_m}, \qquad n_1\le\cdots\le n_m. \] We first obtain an explicit structural decomposition of $\PF_t(G)$ 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 $\PF_t(G)$ is not sequentially Cohen-Macaulay, while for $t\ge 2 n_{m-1}-1$ it is vertex decomposable. Consequently, we obtain a complete characterization for complete multipartite graphs: $\PF_t(G)$ is vertex decomposable if and only if $t\ge 2n_{m-1}-1$. Equivalently, this is also exactly the range in which $\PF_t(G)$ 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.