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I have a conjecture:

"For each planar graph with vertices $V_1, V_2,\ldots, V_n$ there exist disjoint circles $w_1,w_2,\ldots,w_n$ in the plane, such that for every $i,j$, $w_i$ is tangent to $w_j$ if and only if $V_i$ is adjacent to $V_j$."

For example $K_4$ is planar, so we can draw four disjoint and pairwise tangent circles.

My friend told me that this fact has been proved, can anybody help me to find the proof?

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