What would you consider “core genre?” Both in terms of definition and representative titles?
(the genre of the book in question is fantasy, but really I'm more curious how people define any genre - especially the kinds of stories that are "central" vs. along the edges, so answer as your whims take you)
Does this term even make any sense to you?
It makes sense to me, but I've tended to use it and see it used in an unexamined, casual fashion. If I were going to try to unpack the concept, for myself it would go something like this: when someone says "core genre" I think in the context of the entire science fiction and fantasy field (as Jonathan says, different areas within the field can have their own core and periphery), and I think of work that sits in the overlap of several sets: work that is at least modestly commercially successful; work that uses common tropes (work that people unfamiliar with sf would easily recognise as sf); work that follows common stylistic conventions of sf (i.e. is conservative); work that is popular with heavy consumers of sf. The fewer of these circles a work falls into, the less likely it is to be recognised as "core", which is why the concept is so horrendously resistant to evolution. So the work of a writer like Catherynne Valente is not often seen as "core genre", despite selling to New York Times bestseller levels, despite reaching the Hugo ballot, despite critical acclaim, because it is adventurous in style and content. (But Valente would be right in the core of what we might call "literary fantasy", which draws on different sets.) Not unrelatedly, I might also suggest that another set that contributes to the construction of "core genre" is "work produced by people who look like the people who produce sf"; that is, lack of diversity breeds lack of diversity, and when people from diverse backgrounds do start writing sf, they're likely to write sf that challenges the common tropes and styles of the sf that exists, making it harder for them to be recognised as core.
What "core genre" does not mean to me is "good" or "essential"; it is, or at least aspires to be, a historical/commercial judgement, not an evaluative one, and (at least in theory) a group judgement (what we point to when...), not an individual one. Nor is it a stable, definitive assessment. The identification of a core necessarily creates a periphery (or several different ways of being peripheral), and the perception of a core gives the periphery something to push against, leading to exchange between core and periphery over time ... which you could argue is being played out in recent Hugo shortlists. You could also read my award predictions post as being, in part, based on my assessment of the extent to which -- in January 2012 -- each award looks to core or periphery to find its contenders.