Fighting in the country's east has increased since a much-violated 10-day cease-fire was allowed to expire Monday.
President Petro Poroshenko said conditions for further peace talks had not been met as rebels refused to disarm, hand back seized border posts or allow international monitoring.
National Security and Defence Council spokesman Andriy Lysenko said three military personnel died in rebel attacks on government vehicles and checkpoints and 10 were wounded.
The federal border guards said one guardsman was killed when the Novoazovsk crossing point came under attack by rebels with mortars in the Donetsk region.
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Calling Yanukovych's removal a coup, Russia seized and annexed Ukraine's Crimea region in March. Now Ukraine accuses Russia of arming and supporting the separatists in eastern Ukraine, something Russia denies.
Border posts have become a key issue, since Ukraine and the United States say military equipment and reinforcements are flowing across the border from Russia into Ukraine.
Ukraine says it recaptured a key border post yesterday from the rebels at Dovzhanskiy. The guards service said today the insurgents had mined the post with explosives.