US Republican presidential front-runner Donald Trump has called for a "one-on-one" battle with Ted Cruz, urging other rivals to quit the nomination race.
Mr Cruz also suggested it was time for other hopefuls, like Marco Rubio and John Kasich, to step aside.
Mr Trump and Mr Cruz won two states each in Saturday's voting.
In the Democratic race, Bernie Sanders took two states - but Hillary Clinton maintained her front-runner status after a big victory in Louisiana.
Speaking after wins in the Republican Kentucky caucuses and Louisiana primary election, Mr Trump told a news conference: "I would love to take on Ted Cruz one on one."
"Marco Rubio had a very very bad night and personally I call for him to drop out of the race. I think it's time now that he dropped out of the race. I really think so."
Meanwhile, Texas Senator Mr Cruz - who won Republican caucuses in Kansas and Maine - said he believed that "as long as the field remains divided, it gives Donald an advantage".
Florida Senator Mr Rubio and Ohio Governor Mr Kasich have so far made no public comments on the issue.
Mr Cruz now appears to be the only candidate who can stop Mr Trump, analysts say, after a week in which the Republican establishment did everything it could to attack Mr Trump, a New York tycoon.
At a glance:
- Donald Trump won two more states, bringing his total to 12
- Ted Cruz's two victories give him six wins in all
- Marco Rubio has won only one state, while John Kasich has none
- Hillary Clinton's win in Louisiana gives her a total of 11 states
- Bernie Sanders' two victories give him seven wins in all
- Sunday sees further contests in Maine (Democrat) and Puerto Rico (Republican)
Observers also point out that Mr Cruz and Vermont Senator Mr Sanders - who beat Mrs Clinton in Kansas and Nebraska - both won in states holding caucuses but lost in the Louisiana primary elections, involving far more voters.
They say that the Louisiana race also appears to have exposed Mr Sanders' lack of support among African-American voters.
Mrs Clinton said she was thrilled to add to her delegate count.
She said: "No matter who wins this Democratic nomination, I have not the slightest doubt that on our worst day we will be infinitely better than the Republicans on their best day."
Saturday's voting story: By Anthony Zurcher, BBC News
Although Donald Trump has surged to a sizeable lead in the race for the Republican presidential nomination, there were two significant areas in which he had underperformed. He tended to struggle in caucuses and in states where voting is limited to members of the Republican Party.
Saturday's results confirmed that this will continue to be a challenge for the front-runner. He lost closed caucuses to Ted Cruz in Kansas and Maine, and posted a narrow win in Kentucky. He also edged Mr Cruz in Louisiana's closed primary.
Although there aren't many caucuses left on the Republican calendar, most of the forthcoming primaries are closed - including the key vote in Florida in just over a week.
Saturday night's showing helped buttress Mr Cruz's argument that he is the party's best hope to challenge Mr Trump, as Marco Rubio and John Kasich continue to lag far behind. If the Texas senator is to catch the New York billionaire, however, loyal Republican Party voters, particularly in Florida, will have to rally behind him in even greater numbers. It's a tall task - but not an impossible one, if the once-popular Mr Rubio continues to haemorrhage support.
The Democratic results presented yet further evidence that Bernie Sanders can beat Hillary Clinton in smaller, more homogenous states, but she is likely to continue to extend her lead with big wins in Southern and ethnically diverse states, as she did in Louisiana on Saturday.
The election itself, on 8 November, will see America vote for a successor to Barack Obama, a Democratic president standing down after two terms in office which have seen the Republicans take control of both houses of Congress.
Election calendar: Next races
- 6 March: Maine caucuses: Democratic; Puerto Rico primary: Republican
- 8 March: Michigan and Mississippi primaries: Democratic and Republican; Hawaii and Idaho caucuses: Republican
- 15 March: Florida, Illinois, Missouri, North Carolina, Ohio primaries: all Democratic and Republican