In topological graph theory, the Hanani–Tutte theorem is a result on the parity of edge crossings in a graph drawing. It states that every drawing in the...

mid-1930s. Even though Tutte's contributions to graph theory have been influential to modern graph theory and many of his theorems have been used to keep...

eigenvalue of certain Schrödinger operators defined by the graph. The Hanani–Tutte theorem states that a graph is planar if and only if it has a drawing in...

existence theorem for Steiner quadruple systems. He is also known for the Hanani–Tutte theorem on odd crossings in non-planar graphs. Hanani (Chojnacki)...

which is at most equal to the crossing number. However, by the Hanani–Tutte theorem, whenever one of these numbers is zero, they all are. Schaefer (2014...

and the pair-crossing number are not the same. It follows from the Hanani–Tutte theorem that odd-cr(G) = 0 implies cr(G) = 0. It is also known that odd-cr(G) = k...

formula). It also includes the crossing number inequality, and the Hanani–Tutte theorem on the parity of crossings. The second chapter concerns other special...