Colour diversity in spanning structures under Dirac-type conditions

Abstract

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains a Hamilton cycle with $n - O(n^{1/2})$ distinct colours. In the bipartite setting, the analogous question for perfect matchings is closely related to permutations in Latin squares. In this paper, we investigate how a Dirac-type minimum degree condition forces colour diversity in spanning structures. For every constant $1/2 < c \le 1$, we prove the following. $\bullet$ Every properly edge-coloured graph $G$ on $n$ vertices with $δ(G)\ge cn$ contains a Hamilton cycle with at least $cn - O(n^{1/2})$ distinct colours. $\bullet$ Every subset of an $n\times n$ Latin square with at least $cn$ cells in each row and each column contains a permutation with at least $cn - O(n^{2/3})$ distinct symbols. Both bounds are best possible up to the error term.

Theorems & Definitions (26)

• Theorem 1.1: Andersen
• Conjecture 1.2: PY
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 2.1
• Lemma 2.2
• proof