Seymour's self-minor conjecture

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Every countably infinite graph is a proper minor of itself.


Seymour originally conjectured this for any infinite graph, but it turned out to be false [1]. On the other hand, it is known to be true for infinite trees of any size [2]. See also Seymour's self-minor conjecture on the Open Problem Garden.


  1. Oporowski, Bogdan. A counterexample to Seymour's self‐minor conjecture Journal of graph theory 14.5 (1990): 521-524, DOI link
  2. Pott, Julian.The self-minor conjecture for infinite trees. preprint (2009), PDF