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  "body": "\n## Definition\n\nA finite [[graph|00C0BR7D7H]] $G = (V, E)$ is a *penny graph* if there is a map $p \\colon V \\to \\R^2$ such that:\n\n1. $\\norm{p(u) - p(v)} \\ge 1$ for all distinct $u, v \\in V$;\n2. $uv \\in E$ if and only if $\\norm{p(u) - p(v)} = 1$.\n\nEquivalently, $G$ is the contact graph of a family of closed discs of diameter $1$ with pairwise disjoint interiors, centred at the points $p(v)$.\n\n## Remarks\n\nExample: the triangle $K_3$ is a penny graph.\nPut $p$ at the vertices of an equilateral triangle of side $1$; all distances are equal to $1$, so both clauses hold.\n\nNon-example: $K_4$ is not a penny graph.\nThree of its vertices would have to sit at the corners of an equilateral triangle of side $1$.\nNo point of the plane is at distance $1$ from all three corners, so clause 2 fails for the fourth vertex.\n\nEvery penny graph is planar and has maximum degree at most $6$.\nTake a vertex whose point is a corner of the convex hull of $p(V)$.\nIts neighbours are at angles at least $60^\\circ$ apart inside an angle less than $180^\\circ$, so it has degree at most $3$.\nHence every penny graph can be properly coloured with $4$ colours, by colouring greedily after removing such vertices one at a time.\n",
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  "msc": [
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        "what": "an inventory row",
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