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 that in every proper coloring of a k-chromatic graph there is a colorful path Pk on k vertices. If the graph is k-chromatic and triangle-free then in any proper coloring there is also a path Pk 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 Pk. Here we prove the following weaker result providing some evidence towards this conjecture. For a suitable function f(k), in any proper coloring of a {C3, C4}-free f(k)chromatic graph there is an induced colorful path on k vertices. A special case of a result of the first author in [7] says that every triangle-free kchromatic graph G contains an induced path on k vertices. The following more general conjecture is attributed to N.R. Aravind in [2]. A path (or more generally a ∗Research supported in part by grant (no. K104343) from the National Development Agency of Hungary, based on a source from the Research and Technology Innovation Fund. †Research supported in part by grant (no. K104343) from the National Development Agency of Hungary, based on a source from the Research and Technology Innovation Fund.