Combinatorial aspects of colorings on groups
Date
2020
Authors
Singh, Shivani
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Abstract
An r-coloring of a finite group G is any mapping χ∶ G → {1, 2⋯ , r}.
A coloring χ is symmetric if there exists a ∈ G such that, for all x ∈ G,
χ(ax-1 1a) = χ(x). A subset X of a group G is called symmetric if there is
an element g ∈ G, such that gX-1 g = X. We first examine monochromatic
symmetric subsets in r-colorings of finite abelian groups. The combinatorial
aspect of this thesis counts the number of symmetric colorings and equivalence
classes of symmetric colorings of dihedral groups. We also derive polynomials
for the number of symmetric r-colorings of group G × Z2 where G is abelian.
For k, n, r ∈ N, an r-coloring _is said to be k -alternating if every set of k
consecutive vertices have pairwise distinct colors. We calculate the smallest value
of r for which a k -alternating r-coloring of Zn exists. Lastly, we explicitly derive
expressions that count the number of 2-alternating r-colorings and 2-alternating
r-ary necklaces of Zn
Description
A thesis submitted to the Faculty of Science, University of the Witwatersrand
in fulfillment of the degree of Doctor of Philosophy (PhD), 2020