Naegami

Naegami is a slash ship between Byakuya Togami and Makoto Naegi in the Danganronpa series.

Canon
''' WARNING: This section contains heavy spoilers for Danganronpa 1 and Danganronpa 3. '''

The first time they meet each other was during the School Life of Mutual Killing, Byakuya saw Makoto under the impression that he was only an ordinary highschooler unworthy of his time. However, as the story progressed, Byakuya began to admire Makoto's investigative skills, although he would never openly voice it. Byakuya often demanded Makoto accompany him on investigations so he could judge whether Makoto would be a threat to him winning the competition.At the fifth trial Byakuya is surprisingly shown becoming silent and hiding his face  at Makoto apparent sentence,

Which is highly surprising for someone like Byakuya who has been rather untouched by all the events that have surrounded him.

In ultimate despair girls, Byakuya who is working for the future foundation, is shown trying to save Makoto's sister, and at the end of the game he talks to Makoto about how much his prensence has changed him.

In the second game, Makoto, Byakuya, and Kyouko become companions and he supports Makoto's plan to save the remaining students from the will of the Future Foundation's higher-ups. This new bond is reinforced in Danganronpa 3 when Byakuya is willing to come and rescue them after they were forced to take part in the Final Killing Game. He is shown costantly complimenting Makoto. Later, he and his men save Makoto from being gunned down brainwashed Future Foundation soldiers.

Fanon
In some Danganronpa fans' opinion it's possible for Byakuya to have a crush on Makoto, however he would never voice it openly because of his personality (just like it was with his admiration for Makoto's investigative skills), while Makoto's docility towards Byakuya could have meant that was too shy to confess it to him.