Almost all optimally coloured complete graphs contain a rainbow Hamilton path

Hamilton transversals in random Latin squares

“…We note that it was proven by Taranenko [35] (with a simpler proof later found by Glebov and Luria [14]) that $$$ n\times n $$$ Latin squares can have at most $$$ {\left(\left(1+o(1)\right)n/{e}^2\right)}^n $$$ transversals, so that the counting term given in Theorem 1.5 is best possible, up to the exponential error term. Analogously, the authors, together with Kühn and Osthus [15], proved that almost all optimal edge‐colorings (proper edge‐colorings using the minimum possible number of colors) of $$$ {K}_n $$$ admit a rainbow Hamilton path, which proves a stronger statement than Conjecture 1.2 for all but a vanishing proportion of such colorings.…”

“…As the substructures we seek are more complex than intercalates, and we moreover require that they are “well‐spread,” our proof introduces new techniques for switching arguments in Latin rectangles. We note that in [15], the authors, with Kühn and Osthus, used switching arguments to analyze a uniformly random 1‐factorization of $$$ {K}_n $$$ and show that with high probability there is a large collection of subgraphs of a form analogous to that of our $$$ \left(v,c\right) $$$‐absorbing gadgets in the undirected setting. Fortunately, this argument also works in the directed setting with only minor changes that we do not cover here, instead providing the proof of Lemma 5.6 in the appendix of the arXiv version of the paper.…”

“…We remark that this particular absorption strategy, wherein we create an absorbing structure with “flexible” sets, is an instance of the “distributive absorption” method, which was introduced by Montgomery [30] in 2018 and has been found to have several applications since. In particular, this method is also used in [15, 23] to find transversals in random Latin squares and rainbow Hamilton paths in random 1‐factorizations, respectively. Our approach differs from that of [15, 23] in a few key ways.…”

“…In particular, this method is also used in [15, 23] to find transversals in random Latin squares and rainbow Hamilton paths in random 1‐factorizations, respectively. Our approach differs from that of [15, 23] in a few key ways. First, the “asymmetry” of proper $$$ n $$$‐arc‐colorings of $$$ \overleftrightarrow{K_n} $$$ (in comparison to proper edge‐colorings of $$$ {K}_n $$$ with at most $$$ n $$$ colors, which correspond to proper $$$ n $$$‐arc‐colorings of $$$ \overleftrightarrow{K_n} $$$ with monochromatic digons) and “connectedness” of rainbow Hamilton cycles/Hamilton transversals (in comparison to general transversals in Latin squares) necessitate a more complex absorbing structure than the one of either [23] or [15], which is more challenging to create and construct.…”

“…Theorem 1.6 implies that a uniformly random proper $$$ n $$$‐arc‐coloring satisfies Conjecture 1.4 with high probability, which in turn implies that a uniformly random $$$ n\times n $$$ Latin square satisfies Conjectures 1.1 and 1.3 with high probability as well. We note that the number of optimal edge‐colorings of $$$ {K}_n $$$ is a vanishing fraction of the number of $$$ n\times n $$$ Latin squares, so Theorem 1.6 does not imply the result of [15].…”