MathDB

Edges of a planar graph

G

are colored either with blue or red. Prove that there is a vertex like

v

such that when we go around

v

through a complete cycle, edges with the endpoint at

v

change their color at most two times.
Clarifications for complete cycle: If all the edges with one endpoint at

v

are

(v,u_1),(v,u_2),\ldots,(v,u_k)

such that

u_1,u_2,\ldots,u_k

are clockwise with respect to

v

then in the sequence of

(v,u_1),(v,u_2),\ldots,(v,u_k),(v,u_1)

there are at most two

j

such that colours of

(v,u_j),(v,u_{j+1})

(

j \mod k

) differ.

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