Proper Hamiltonian Paths in Edge-Coloured Multigraphs
  Given a $c$-edge-coloured multigraph, a proper Hamiltonian path is a path that contains all the vertices of the multigraph such that no two adjacent edges have the same colour. In this work we establish sufficient conditions for an edge-coloured multigraph to guarantee the existence of a proper Hamiltonian path, involving various parameters as the number of edges, the number of colours, the rainbow degree and the connectivity. 
Raquel \'Agueda, Valentin Borozan, Marina Groshaus, Yannis   Manoussakis, Gervais Mendy, Leandro Montero
cs.DM math.CO