The tanks entered Ukraine along with other armour through a checkpoint controlled by rebels in the Luhansk region, Arsen Avakov said.
The army engaged two of them and destroyed part of the column, he said.
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Meanwhile, the Russian and Ukrainian presidents have discussed plans to defuse the crisis, reports say.
Vladimir Putin and Petro Poroshenko, who was elected last month, exchanged their views in a telephone conversation, Kremlin spokesman Dmitry Peskov was quoted as saying by Russia’s media.
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