From chapter two of The Hobbit, Roast Mutton:
They have seldom even heard of the king round here...
Which king are the dwarves referring to?
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I don't know if Tolkien had a specific king in mind when he first wrote that sentence — if it was present in the first revision, this would have been some abstract, distant king. (The Hobbit was first written as a standalone story, and then revised to fit in the world of The Lord of the Rings.)
In the LOTR continuity, the Shire is part of the kingdom of Arthedain, which is one of the parts of Arnor. Although the kingdom of Arthedain effectively fell about a thousand years before the events in The Hobbit, the Shire was still nominally under the rule of the (vacant) throne of Arnor; the Thain, in theory, held his authority in the king's name. When Aragorn restored the joint kingdom of Arnor and Gondor, he was the de jure king of the Shire as well as the rest of Eriador.
This is one of many issues raised by The Hobbit that Tolkien anticipated and covered in the Prologue to The Lord of the Rings. In the third paragraph of the section "Of the ordering of the Shire" he says: "there had been no king for nearly a thousand years, and even the ruins of Kings' Norbury were covered with grass. Yet the Hobbits still said of wild folk and wicked things (such as trolls) that they had not heard of the king."
Even though the expression is attributed to the hobbits, not to dwarves, this is obviously meant as a gloss on its use in The Hobbit. The text there reads: "Others said: '. . . They have seldom even heard of the king round here'". Unless we believe that a group of dwarves said the line in chorus, it must be the diarist -- Bilbo -- who has summed up one of the views voiced in the travellers' debate in these words.
This is what my husband, the walking Tolkien encyclopedia, said when I asked:
No, @Gilles just about covered it. I would have parenthetically fleshed out the definition of “the Thain” (basically a Hobbit steward of the Shire who rules in the King’s name). But the bottom line is “the king” is the king of Gondor, Arnor, and Eriador. At the time of the Hobbit, there was no actual King (the line had died out and the Stewards ruled Gondor and basically ignored the kingdoms of the North), but that’s a technicality that would be lost on the provincial Hobbits—“the king” is as good a vernacular term as any.