The Houthis have stepped up attacks on Saudi border posts in recent weeks, despite three months of bombardment by Saudi-led coalition intent on restoring Yemen's government in exile.
Read: Two Saudi border guards die in Yemen shelling
Two soldiers from the Royal Saudi Land Forces, one from the kingdom's Border Guards, and a UAE officer were killed on Wednesday in three incidents along the border, a mostly mountainous area.
The Houthis and their allies in Yemen's army have managed to hold most of the populated western part of the country against armed groups in Aden, Taiz, Dhalea and Marib backed by coalition air strikes.
But a spate of bombings in the Houthi-controlled capital continued on Thursday, as a remote controlled bomb detonated outside the state news agency according to residents, killing a Houthi fighter and wounding two others.
Read: Escalating conflict: Houthis seize key provincial capital near Saudi border
Saudi Arabia's coalition of Arab countries wants to force the Houthis and Saleh to quit captured areas, return seized arms and let President Abd-Rabbu Mansour Hadi return from his Riyadh exile in line with UN Security Council resolution 2216.
However, neither side appears ready to make concessions, causing the collapse of UN-sponsored talks last week. Aid groups say Yemen's humanitarian crisis is worsening as a result of fighting and a coalition embargo.
Saudi Arabia believes the Houthis are closely tied to the kingdom's main regional foe, Iran, though diplomats in the region say those ties may be exaggerated.
The Houthis say they are winning a revolution against corrupt officials and Sunni Muslim militants they say are backed by Saudi Arabia and the exiled government.
A series of car bombs against mosques close to the Houthi movement last week was claimed by Islamic State, in a sign that political chaos was giving militant groups more room to operate.
Read: Saudi-led warplanes bomb rebels across Yemen
Al Qaeda in the Arabian Peninsula, the strongest branch of the global militant group, carries weapons and patrols openly in the eastern city of Mukalla but a wave of suspected US drone strikes have killed its top leaders in recent weeks, including its chief Nasser al-Wuhayshi – al Qaeda's deputy leader.
Residents reported another suspected drone strike on Wednesday which killed four militants inside an army base near Mukalla that had been taken over by al Qaeda.
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