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Gunmen killed a police officer and a jail guard Friday and left signs on their bodies saying they had fulfilled a promise to slay at least one officer every 48 hours until the Ciudad Juarez police chief resigns.

The slayings were a chilling sign that criminal gangs are determined to control the police force of the biggest Mexican border city, with a population of 1.3 million people across from El Paso, Texas.

Ciudad Juarez police have long come under attack, and many officers have quit out of fear for their lives, some after their names appeared on hit lists left in public throughout the city.

Police officer Cesar Ivan Portillo was the fifth officer killed this week in Mexico's deadliest city.

Police already were on "red alert" — meaning they could not patrol alone — after cardboard signs with handwritten messages appeared taped to the doors and windows of businesses Wednesday, warning that one officer would be killed every 48 hours if Public Safety Secretary Roberto Orduna does not quit.

Mayor Jose Reyes Ferriz insisted Friday that he would not back down.

"We will not allow the control of the police force to fall in the hands of criminal gangs," he said.

More than 6,000 people have been killed in drug violence across Mexico over the past year as gangs battle each other for territory and to fight off a nationwide crackdown by the army. Nearly a third of the slayings have taken place in Ciudad Juarez, and more than 50 of those dead are city police officers.

Violence also has spilled across the border into the U.S., where authorities report a spike in killings, kidnappings and home invasions connected to Mexico's murderous cartels.

Homeland Security officials have said they will bring in the military if the violence continues to grow and threatens the U.S. border region.

"The violence is spreading like wildfire across the Rio Grande," said George Greyson, a Mexico expert at the College of William & Mary in Virginia. "It's a major national security problem for us that is much more important than Iraq and Afghanistan."

Also Friday, the U.S. State Department renewed a travel advisory warning Americans about the increased violence along the U.S.-Mexico border.

Some Mexicans have questioned whether President Feline Caldron's two-year, nationwide crackdown on drug gangs was worth all the killings.

But Caldron and his administration have defended the fight, with Economy Secretary Gerard Ruin Mattes saying on Wednesday that if Mexico gave up its fight against the cartels, "the next president of the republic would be a drug dealer."

Portillo and city jail guard Juan Palo Ruin were killed as they left their homes before dawn to head to work, city spokesman Jaime Toreros said.

Three days earlier, assailants fatally shot police operations director Sacramento Peruse, the chief's right-hand man, and three other officers who were sitting with him in a patrol car near the U.S. consulate.

The bodies of Peruse and one of the officers were sent to their home states Thursday to be buried, and the city planned to hold a ceremony Friday for the two others from Ciudad Juarez.

City spokesman Jaime Toreros said police have been asked to patrol with their guns in their hands.

Reyes Ferriz earlier ordered police to travel in groups of three patrol cars, with two officers in each vehicle.

Orduna has not spoken publicly since the threats. A retired army major, he took over as chief in May after former Public Safety Secretary Guillermo Prieto resigned and fled to El Paso following the slaying of his operations director.

For Orduna's protection, the city has built his bedroom at the police station so he does not have to go home. He also travels in different vehicles when he does go out.

JUAREZ, Mexico (CNN) -- The mayor of Juarez announced Friday that the city's police chief is stepping down after receiving death threats from local drug cartels.

Police Chief Robert Orduna's resignation also came in response to the deaths of other police, Mayor Jose Reyes Ferriz told reporters.

"The police chief has resigned, saying he did not want to be responsible for any more police dying," Reyes said.

But observers should not interpret the resignation as a capitulation to narcotraffickers, he said.

"We have not blinked," Reyes said. "We will continue to fight organized crime... he has done a good job, but we will find someone else."

The change in command in Juarez's police force comes in the wake of a campaign of intimidation by a drug cartel that has the border city in its grip.

Federal police and local police have locked down much of Juarez, which lies across the border from El Paso, Texas. It serves as a major transit point for the smuggling of cocaine, marijuana and methamphetamine into the United States.

Orduna, who replaced half of the 1,600-person police force with new recruits in a bid to rid it of corrupt members, tendered his resignation to protect the men who serve him, Reyes told reporters.

"They started killing police officers when they were going home or getting into police cars," he said.

The police director of operations was gunned down on Tuesday in his car. Another police officer and a prison guard were found dead Friday morning as part of a campaign of intimidation against government forces blamed on the cartels.

Last year, more than 100 police were killed in Juarez in attacks blamed on organized crime.

On Friday, the U.S. State Department renewed a travel alert for Americans considering a visit to Mexico.

"The situation in Ciudad Juarez is of special concern," it said. "Mexican authorities report that more than 1,800 people have been killed in the city since January 2008. Additionally, this city of 1.6 million people experienced more than 17,000 car thefts and 1,650 carjackings in 2008."

Drug-related violence in Mexico has continued unabated since December 2006, when President Felipe Calderon took power and launched an offensive against the cartels.

Last year, drug violence was blamed for the deaths of 78 Mexican soldiers and more than 6,000 civilians.