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  "body": "\n## Definition\n\nLet $G$ and $H$ be [[graphs|0JD5C82QMB]] with vertex sets $V(G)$ and $V(H)$.\nA graph homomorphism from $G$ to $H$ is a map $f : V(G) \\to V(H)$ such that $f(u) \\sim f(v)$ in $H$ whenever $u \\sim v$ in $G$.\n\n## Remarks\n\nA homomorphism maps edges to edges. It may map non-adjacent vertices to adjacent ones, and it may map several vertices to one vertex.\nA homomorphism from $G$ to the complete graph $K_k$ is the same as a proper [[colouring|0874138YP6]] of $G$ with $k$ colours.\nA bijective homomorphism whose inverse is also a homomorphism is a [[graph isomorphism|00CT0PTH74]].\n\n## Explanations\n\n### L1 human\n\nA graph homomorphism is a map of vertices that sends each edge to an edge.\nFor example, mapping the vertices of a square alternately to the two ends of one edge is a homomorphism from $C_4$ to $K_2$.\n",
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