Combinatorial aspects of colorings on groups

Singh, Shivani
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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
A thesis submitted to the Faculty of Science, University of the Witwatersrand in fulfillment of the degree of Doctor of Philosophy (PhD), 2020