Proper Hamiltonian Cycles in Edge-Colored Multigraphs
  A $c$-edge-colored multigraph has each edge colored with one of the $c$ available colors and no two parallel edges have the same color. A proper Hamiltonian cycle is a cycle containing all the vertices of the multigraph such that no two adjacent edges have the same color. In this work we establish sufficient conditions for a multigraph to have a proper Hamiltonian cycle, depending on several parameters such as the number of edges and the rainbow degree. 
Raquel \'Agueda, Valentin Borozan, Raquel D\'iaz, Yannis Manoussakis,   Leandro Montero
cs.DM math.CO