Properly-Coloured Hamilton Cycles and the Bollobás-Erdős Conjecture

This lecture explores the conditions under which edge colorings of complete graphs must contain properly-colored Hamilton cycles, where no adjacent edges share the same color. Delve into the famous 1976 Bollobás-Erdős conjecture, which states that if every vertex is adjacent to fewer than ⌊n/2⌋ edges of the same color in a complete graph with n vertices, then a properly-colored Hamilton cycle must exist. Examine the tightness of this bound through various extremal examples presented by Bollobás and Erdős, Fujita and Magnant, and Lo. Learn about recent developments including a proof of this conjecture for large n, presented by Richard Montgomery, an award-winning mathematician from the University of Warwick whose research in extremal and probabilistic combinatorics has earned him the European Prize in Combinatorics, a Philip Leverhulme Prize, and an European Mathematical Society prize.