$($a$)$

$($b$)$

In Figure $($b$),$ can the agent color the map using three colors if two neighbor states

cannot be colored the same? Please prove it logically and mathematically.

NOTE: $(1)$ assume state $3$ and state $5$ have a short boundary $($i$.$e$.,$ they neighbors$),$

and state $2$ and state $7$ are not neighbors; $(2)$ the proof doesn't count if it is done by

running the program and examining the results.