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The pseudoachromatic number ψ(G) of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that between any two distinct parts, there is at least one edge of G. Here, we prove that if 2⩽a⩽b⩽c, then there exists a graph G with chromatic number a, achromatic number b and pseudoachromatic number c.
The pseudoachromatic number of a graph is the largest number of colours in a (not necessarily proper) vertex colouring of the graph such that every pair of distinct colours appears on the endpoints of some edge. The achromatic number is largest number of colours which can be used if the colouring must also be proper.Hedetniemi (http://cyclone.cs.clemson.edu/hedet/coloring.html) conjectured that these...
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