Monogami

Monogami is the slash ship between Monokuma and Byakuya Togami from the Danganronpa fandom.

Danganronpa: Trigger Happy Havoc
While Byakuya Togami hates Monokuma and swears to take him and his mastermind down, he and Monokuma have some shared ideals and occasionally work together due to the fact that Byakuya, in the beginning of the game, found the killing game to be enjoyable. Monokuma was entertained by the fact that Byakuya was enjoying the game and became supportive of his determination, occasionally helping him and referring to him with greater respect than the other students.

The best example of this is in Chapter 2, where Byakuya decides to alter the crime scene of the victim in order to expose Toko's identity as Genocide Jack. During the investigation, Byakuya misleads the other characters into thinking that Genocide Jack was the killer. At one point he even claps his hands together and summons Monokuma who immediately shows up, to which Makoto internally questions whether or not Monokuma has been domesticated by Byakuya. Monokuma tells Makoto to get his poop together and be more like Byakuya after being tired of his questioning. During the trial, Monokuma suggests that the two of theme could potentially be friends due to their shared love of complicating things, leading Byakuya to refer to him as a lowly criminal who is beneath him. Monokuma responds to him by calling him grumpy.

In Chapter 4, Byakuya claims that Yasuhiro is low on the totem pole of life when uninterested in recieving Monokuma's motive. Later after the chapter's trial, Byakuya agrees with Monokuma who claims that the killing game is a competition between the students. After he agrees, Monokuma refers to Byakuya as "lovely Byakuya" and claims that he understands, to which Byakuya then says he's dropping out of the killing game and going directly for the mastermind with the others. Monokuma is enraged by his statement.

Monokuma's one-sided respect for Byakuya continues into Chapter 5 and 6, however. Monokuma occasionally shows up to answer questions for Byakuya during investigations and give him details. During the 5th trial, when Byakuya asks Monokuma to shut up, he responds with "Shutting up, boss!" and immediately goes silent upon Byakuya's request. In Chapter 6, Byakuya finds an inactive Monokuma unit and he along with the others take it apart to inspect it. He learns at the end that Monokuma has been controlled remotely by Junko Enoshima, making Monokuma's actions towards him simply an act by her.

Danganronpa: The Animation
Similarly to Trigger Happy Havoc, Monokuma shows some respect and gets some enjoyment out of Byakuya's playing along with the killing game. However, Byakuya still shows an intense hatred towards him. In the anime, Monokuma seems to either be afraid or is acting afraid of Byakuya, shown during Episode 11 where, when giving out a class photo to Byakuya, he begins sweating profusely and runs away the moment that Byakuya notices him.

School Mode
The two don't really interact during school mode, but when Makoto makes note of the Monokuma merchandise in the school store, Byakuya is immediately disgusted and says to get it away from him, leading in a bad outcome. It should be noted that school mode is not canon.

Danganronpa Another Episode: Ultra Despair Girls
While Monokuma doesn't directly show up as a character in Ultra Despair Girls, Byakuya is shown to be shooting down the Monokuma units with a hacking gun. Upon meeting Komaru, he asks why the "vermin Monokuma" are everywhere and is disgusted by them.

Danganronpa S: Ultimate Summer Camp
In the non-canon switch board game Danganronpa S: Ultimate Summer Camp, Monokuma is his own sentient AI who functions separately from any of the canonical masterminds. During his Potential for Talent event, one of the options is for Monokuma to build his school anew. He goes to Byakuya and asks him if he would like to invest in him, explaning that Byakuya would have nothing to lose from working with him and that he would erect a statue of him. He also states that he would have the students wear Byakuya masks. When Byakuya is enraged and tells him never to talk to him again, Monokuma tells him off, prompting him to leave. Once Byakuya leaves, Monokuma seems to be temporarily distraught, but then claims that he wanted to give him priority entrance to the next killing game, implying that Monokuma wasn't really valuing him.

Quotes
"Be more like Byakuya and get your poop together." – Monokuma to Makoto during the Chapter 2 investigation.

Monokuma – "Ahh, the noble son of a noble family...Truly you understand me! Puhuhu, I think this is the start of a terrifying friendship!" Byakuya – "Shut up. I would never stoop to the level of a childish criminal like you." – Monokuma and Byakuya after Byakuya complicates the Chapter 2 trial.

"Ahh, lovely Byakuya. You understand, don't you?" – Monokuma referring to Byakuya in Chapter 5 when Byakuya momentary agrees with him on the purpose of the killing game.

Byakuya – "Shut up, you!!" Monokuma – "Shutting up, boss!" – Byakuya and Monokuma during the Chapter 5 trial.

Monokuma – "How cold. You have nothing to lose if you're on Team Monokuma! Once my school is done, I'll erect a statue of you! I'll even have every student wear a Byakuya mask!" – Monokuma's Development Plan in Danganronpa S: Ultimate Summer Camp.

Fanon
The ship is considered to be a crackship and stemmed from fans mostly as a joke, with various reddit, amino, youtube and tumblr posts doing edits or drawings of the two together. Because of Monokuma's oddly respectful treatment of Byakuya, and Byakuya's interest in making the Killing School Life more exciting, the two have an occasionally mutually supportive dynamic which may have fueled the ship, especially with Monokuma's use of the term "lovely Byakuya" and the fact that he showed up on command when Byakuya clapped in Chapter 2.

Due to the V3 existence of Monosuke, the monokub, who is fond of money and has a yellow/blonde color scheme like Byakuya, some fans have joked that Monosuke is actually Byakuya and Monokuma's son. While a few fans do ship the bear and Byakuya seriously, it is mostly written and portrayed as something silly. In some game grumps thumbnails of their playthrough of Trigger Happy Havoc, Monokuma is shown with hearts while next to Byakuya.

Some fans draw Monokuma as a human, pairing the two together as a serious ship where Monokuma either leaves Junko as mastermind and joins Byakuya Togami instead. There is a number of fanart of the two together. Because Monokuma is canonically a bear and enjoys being one, this is considered more of an OC x Canon ship by the fandom, though some consider it to be canon x canon.

Because Junko controls Monokuma and Junko reveals at the end of Chapter 6 that she wiped out the entire Togami lineage, some fans find the ship to be toxic if taken seriously. Many see the ship to technically be EnoGami (Junko Enoshima x Byakuya Togami) due to the fact that Monokuma is one of Junko's personalities. Many fans also find the ship to be uncomfortable due to the fact that it would be an interspecies relationship between a robotic bear and a human.

The ship isn't very well known, being nowhere near as popular as ships such as Naegami and Monokumi.

FAN FICTION
Monokuma/Byakuya tag on AO3

AMINO
Togakuma Amino