The Chromatic Number of the Two-Packing of a Forest

Summary

A two-packing of a graph G is a bijection σ : V (G)↦V (G) such that for every two adjacent vertices a, b ∈ V (G) the vertices σ(a) and σ(b) are not adjacent. It is known [2, 6] that every forest G which is not a star has a two-packing σ. If F _σ is the graph whose vertices are the vertices of G and in which two vertices a, b are adjacent if and only if a, b or σ ^{− 1}(a), σ ^{− 1}(b) are adjacent in G then it is easy to see that the chromatic number of F _σ is either 1, 2, 3 or 4. We characterize, for each number n between one and four all forests F which have a two-packing σ such that F _σ has chromatic number n.