Only-C chain molecules
The term you are looking for is "allotropes", and carbon has a whole bunch of them, but the three most readily spotted in nature are graphite, diamond and fullerene, however, none of them are simple chains such as those postulated in the answer.

Graphite exists in large honeycomb flakes held together on two axes by covalent bonds, and on the third by weaker Van der Waals forces, which is why they come off readily under mechanical shear, leading to their popularity in pencil lead.

Diamond are huge, multiply interlocked complexes of crystalline carbon, although often with impurities locked into their structure, perverting the crystal structure and yielding interesting colors as a byproduct.

Fullerene is a molecular allotrope, consisting (usually) of 80 carbon atoms arranged in a buckyball. Initially thought to be a lab-grown species, it has since been identified as a natural result of certain combustion processes.

Now, none of these are chains like what you describe; while some of the compounds you sketch out are likely possible and present, they are also likely to be highly reactive, perhaps even to the point of decomposing spontaneously.

While amorphous carbon chains may very well exist, without a crystalline structure to shield their electrons, their reactivity would likely be such that they would be unable to exist for very long at all.

Simplistically speaking, the long stable chains of carbon found in hydrocarbons aren't entirely characteristic of carbon, as such, but rather, characteristic of hydrogenated carbon.