Comment on 'properly Coloured Hamiltonian Paths in Edge-coloured Complete Graphs without Monochromatic Triangles'

摘要

Fix a positive integer c. Let K_n~φ denote the complete graph K_n of order n whose edges have been colored by φ:E(K_n) → C with #C = c. A subgraph H of K_n~φ is properly colored if any two adjacent edges in H have distinct colors; and, is monochromatic if all edges in H have the same color. Olof Barr [1] recently proved the foloowing result (here c ≥ 3).