Let G be a graph. The path graph P3(G) of G is obtained by representing the paths P3 in G by vertices and joining two vertices whenever the corresponding paths P3 in G form a path P4 or a cycle C3. In this paper

we show that for a triangle-free graph G

χ(P3(G))≤β(G)

where β(G) is the vertex covering number of G. For a connected graph G of order at least 3

χ(P3(G))≤2 if and only if G is bipartite. Furthermore

χ(P3(G))=1 if and only if G is a star. For a K4-subdivision graph G

2≤χ(P3(G))≤3. For a series-parallel graph or an outerplanar graph G

χ(P3(G))≤3.