These two pieces by Borges are among his briefest and most iconic. The parable of two kings is a masterful example of building an entire labyrinth with a scarce few sentences, while his reflection on dreamtigers is a prime example of the surrealist tradition.
The story of the two kings and two labyrinths tells of a king who, in the first age of mankind, ruled the isles of Babylonia. He ordered his architects and priests to build a labyrinth in which even the most prudent men would be lost (263). His goal was to create fear and awe with his labyrinth, and the story suggests he overstepped the domain of humanity in so doing.
Eventually, an encounter transpired when an Arab king came to visit. The Babylonian king wanted to make fun of the Arab king's simplicity, and so invited him into the labyrinth. The Arab king obliged, and wandered until evening came, when he implored the help of God and promptly located the door.
Though the Arab king did not complain, he told the Babylonian king that he also had a labyrinth which he would ensure the Babylonian king one day saw. In keeping with this promise, after returning home, the Arab king ransacked Babylonia with his men and captured their king. They rode with him for three days and three nights into the desert, where the Arab king left him.
Before leaving him to die, the Arab king tells him that this is his labyrinth, where, in contrast with the Babylonian king's maze, God has seen fit to remove all obstacles and impediments to passage. With no food or water, the king soon perishes. The tale ends with the invocation "Glory to him who does not die" (264).
In Dreamtigers, Borges describes his childhood love for tigers - specifically, the sort which would ride majestically in castles atop elephants. He studied them at the zoo and in books, and revels in the fact that he can still perfectly remember their pictures, when he cannot properly muster up in his mind the faces of women (294). He is grateful that the tigers are still able to come to him in his dreams.
Yet his dreams are also cause for lamentation, because when he is aware that he is dreaming and chooses to exert his will by summoning a tiger, he is not able to make the desired creature appear. Something will arise, but it will always have issues: "it is all dried up, or it's flimsy-looking, or it has impure vagaries of shape or an unacceptable size, or it's altogether too ephemeral, or it looks more like a dog or bird than like a tiger" (294).
The two labyrinths ought to call to mind Tlön, Uqbar, Orbis Tertius and The Lottery: one way to think of them is as symbols for institutionalized religion which passes itself on as divine in the first labyrinth, versus the truly divine God expressed in the infinity of the second labyrinth. Recall that the Lottery described a human institution which sought to package a phenomenological metaphysical concept - chance - and sell it back to society wholesale. This is a similar situation: the Babylonian king is trying to engineer the infinite, but only the universe itself is infinite, being the true labyrinth.
This also constitutes a classic case of hubris. The Babylonian king wanted to prove his ultimate superiority by achieving the divine in a labyrinth and humiliating another king by trapping him within. Yet he was blinded, and in his blindness was punished by the king who was an ardent adherent to the true nature of the cosmos, invoking it to escape from the artifice of the man-made labyrinth.
Dreamtigers hearkens back to The Circular Ruins, addressing themes of liminality and the difference between dreams and reality. The crux of the anecdote can be found in the fact that although Borges can remember the tigers perfectly in wakefulness, he cannot faithfully recreate them when he attempts to will them into being in his dreams. The awareness that it is a dream is key: because he is aware that he is in a dream, he cannot be satisfied with a tiger that he sees - he knows it is not real.
Compare Borges in this story to the dreamer in The Circular Ruins. The efforts of the dreamer were far more successful than Borges, but the dreamer was also unaware that he was a dream. Borges, aware of being in a dream, cannot create anything well. Taken as a pair, these stories suggest something novel and compelling about the liminality between dream and reality: the liminality is purely a mental barrier.