Chromatic Number of Graphs each Path of which is 3-colourable

In his paper “Fruit salad” (mixed for Paul Erdos) Gyárfás has posed the following conjecture: If each path of a graph spans at most 3-chromatic subgraph then the graph is k-colourable (with a constant k, perhaps with k = 4). We will show that these graphs are colourable with 3 · Illg c ¦V(G)¦⌉ colours for a suitable constant c = 8/7. As a corollary we obtain that every graph G admits a partition of its vertex set V(G) into at most Illg c ¦V(G)¦⌋ subsets V i for a suitable constant c = 8/7, such that the components of each induced subgraph G[V i ] are spaned by a path.