2016/04/20 by Babu, Jasine, Basavaraju, Manu, Chandran, L. Sunil +1
#05C15 #Combinatorics (math.CO) #FOS: Mathematics #G.2.2
paper · doi:10.48550/arxiv.1604.06070
Given a graph G=(V,E) whose vertices have been properly coloured, we say that a path in G is "colourful" if no two vertices in the path have the same colour. It is a corollary of the Gallai-Roy-Vitaver Theorem that every properly coloured graph contains a colourful path on χ(G) vertices. We explore a conjecture that states that every properly coloured triangle-free graph G contains an induced colourful path on χ(G) vertices and prove its correctness when the girth of G is at least χ(G). Recent work on this conjecture by Gyárfás and Sárközy, and Scott and Seymour has shown the existence of a function f such that if χ(G)≥ f(k), then an induced colourful path on k vertices is guaranteed to exist in any properly coloured triangle-free graph G.