Induced Colorful Trees and Paths in Large Chromatic Graphs
In a proper vertex coloring of a graph a subgraph is colorful if its vertices are colored with different colors. It is well-known (see for example in Gyárfás (1980)) that in every proper coloring of a $k$-chromatic graph there is a colorful path $P_k$ on $k$ vertices. The first author proved in 1987 that $k$-chromatic and triangle-free graphs have a path $P_k$ which is an induced subgraph. N.R. Aravind conjectured that these results can be put together: in every proper coloring of a $k$-chromatic triangle-free graph, there is an induced colorful $P_k$. Here we prove the following weaker result providing some evidence towards this conjecture: For a suitable function $f(k)$, in any proper coloring of an $f(k)$-chromatic graph of girth at least five, there is an induced colorful path on $k$ vertices.