Near Rainbow Hamilton Cycles in Dense Graphs

ABSTRACT The problem of finding near‐rainbow Hamilton cycles in properly edge‐coloured graphs was first studied by Andersen, who proved in 1989 that every proper edge colouring of the complete graph on vertices contains a Hamilton cycle with at least distinct colours. This result was improved to by Balogh and Molla in 2019. In this paper, we consider Anderson's problem for general graphs with a given minimum degree. We prove every globally ‐bounded (i.e. every colour is assigned to at most edges) properly edge‐coloured graph with contains a Hamilton cycle with distinct colours. Moreover, we show that the constant 1/8 is best possible.