The chromatic number of the two-packing of a forest

Abstract

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.