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arXiv · 1509.03855

Homotopy types of Hom complexes of graphs

Takahiro Matsushita
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Abstract

The Hom complex ${\rm Hom}(T,G)$ of graphs is a CW-complex associated to a pair of graphs $T$ and $G$, considered in the graph coloring problem. It is known that certain homotopy invariants of ${\rm Hom}(T,G)$ give lower bounds for the chromatic number of $G$. For a fixed finite graph $T$, we show that there is no homotopy invariant of ${\rm Hom}(T,G)$ which gives an upper bound for the chromatic number of $G$. More precisely, for a non-bipartite graph $G$, we construct a graph $H$ such that ${\rm Hom}(T,G)$ and ${\rm Hom}(T,H)$ are homotopy equivalent but $\chi(H)$ is much larger than $\chi(G)$. The equivariant homotopy type of ${\rm Hom}(T,G)$ is also considered.

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Takahiro Matsushita. 2015-09-13. Homotopy types of Hom complexes of graphs. https://arxiv.org/abs/1509.03855

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