The Chromatic Number of the Two-packing of a Forest
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 ,  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.
KeywordsPacking Placement Factorization Tree Forest Chromatic number
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