Igor Karkaroff's relationships are fascinating because they're almost entirely transactional and marked by profound cowardice. His connection to Snape is the most layered—they were Death Eaters together, but Snape's defection and subsequent protection from Dumbledore must have chafed Karkaroff enormously. Karkaroff tries to use their shared past as leverage, dropping hints to Dumbledore at the Yule Ball, but Snape just stares him down. It's a dynamic built on mutual recognition of past sins, but zero mutual respect.
Then there's Dumbledore. Karkaroff is terrified of him, not because Dumbledore is a threat in the same way Voldemort is, but because Dumbledore sees right through his act. Dumbledore's polite disdain at the Triwizard arrival speaks volumes. Karkaroff's relationship with his own student, Viktor Krum, also feels utilitarian; he's a tool for Hogwarts glory, not a young man Karkaroff genuinely mentors. His panicked, final appearance, trying to sell out fellow Death Eaters to save his own skin, perfectly crystallizes every relationship he has—they're all just currency for his survival.