The 13th, 14th, and 15th Amendments all go together in a lot of ways. The 13th Amendment made slavery illegal, the 14th Amendment granted former slaves United States citizenship, and the 15th Amendment gave former slaves and African American men the right to vote.
The 14th Amendment gave former slaves United States citizenship, but it also protected them under "equal protection of the laws." This meant that they would have to be treated equally, which didn't always end up happening. Many people believed that segregation on buses violated the 14th Amendment, as they weren't being treated equally.
On June 5, 1956, bus segregation was ruled to be unconstitutional and it was said to have violated the 14th Amendment. This was the same with the Brown v. Board of Education case, which ended up getting rid of segregation in American public schools.