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Addendum to: Aaron Archer ``On the upper chromatic numbers of the reals,''

Summary: Addendum to:
Aaron Archer ``On the upper chromatic numbers of the reals,''
Discrete Math., 214 (2000) 65­75.
In the journal paper, I included the following theorem without proof, since the proof is very similar to
Erdos's probabilistic proof that there exist graphs with arbitrarily high girth and chromatic number (as it
appears in [1, p. 35]). Since some people have inquired about the proof, I set it down here for completeness.
Theorem 14. Given any l, m # N there exists a graph G with girth(G) > l and # cap (G) # m.
Proof: Given a graph G = (V, E) with its edges colored from the set [m], let E i denote the set of edges
that are colored i, and let G i = (V, E i ), i.e., the subgraph of G induced by the edges colored i. Let #(G i )
denote the independence number of G i , in other words, the maximum number of vertices of G that can be
colored i in a vertex coloring compatible with G's edge coloring.
We will show using the probabilistic method that there exists a simple edge­colored graph G = (V, E)
with a small number of cycles of length at most l, such that #(G i ) is small for each i. We then delete an
arbitrary edge from each such short cycle, yielding a new graph G # . By construction, girth(G # ) > l. Let D
denote the set of deleted edges. Since #(G # i ) # #(G i )+|D#E i |, # i #(G # i ) is still small. Choosing appropriate
parameters of our probabilistic construction, we will get # i #(G # i ) < n (where n = |V |), which implies that
G # has no compatible vertex coloring.
Let # < 1
l , p = mn #-1 . Let G be a random graph on n vertices, where each of the # n
2 # possible edges


Source: Archer, Aaron - Algorithms and Optimization Group, AT&T Labs-Research


Collections: Computer Technologies and Information Sciences