2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67527For any two graphs $G$ and $H$ Lovász has defined a cell complex $Hom(G,H)$ having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with $G$ a cycle of odd length. More specifically, we show that: if $Hom(C_{2r+1},G)$ is $k$-connected, then $χ(G)\geq k+4$. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney classes.This is a research announcement, which is to appear in ERA-AMSCombinatoricsAlgebraic Topology05C15; 57M15, 55N91, 55T99text