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2026-03-26 19:03:08.1774551788
Prove that a bipartite planar graph has a vertex of degree at most 3
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If there are at most $3$ vertices in the graph, the degree of a vertex is at most $2$.
This answer shows that for a planar bipartite graph with $v > 3$ we have
$$ e \leq 2v - 4 \enspace, ~~~~~~~~~~~~~~~~~~(*) $$
where $v$ is the number of vertices and $e$ is the number of edges. The handshaking lemma says the sum of the degrees of all vertices is $2e$. Hence if all vertices have degree at least $4$, $2e \geq 4v$, which contradicts $(*)$.