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We prove that unless Exponential Time Hypothesis (ETH) fails, deciding ifthere is a homomorphism from graph $G$ to graph $H$ cannot be done in time$|V(H)|^{o(|V(G)|)}$. Combined with the reduction of Cygan, Pachocki, andSoca{\l}a, our result rules out (subject to ETH) a possibility of$|V(G)|^{o(|V(G)|)}$-time algorithm deciding if graph $H$ is a subgraph of $G$.For both problems our lower bounds asymptotically match the running time ofbrute-force algorithms trying all possible mappings of one graph into another.Thus, our work closes the gap in the known complexity of these fundamentalproblems.

Tight Bounds for Subgraph Isomorphism and Graph Homomorphism