DMS Combinatorics Seminar

Speaker: Stacie Baumann

Title: Embedding and Coloring Designs

Abstract: This presentation focuses on the two problems contained in my dissertation.

The main topic of the presentation is the first problem that concerns completing partial latin squares with prescribed diagonals. Necessary and sufficient numerical conditions are known for the embedding of an incomplete latin square $L$ of order $n$ into a latin square $T$ of order $t ≥ 2n + 1$ in which each symbol is prescribed to occur in a given number of cells on the diagonal of $T$ outside of $L$. This includes the classic case where $T$ is required to be idempotent. If $t < 2n$ then no such numerical sufficient conditions exist since it is known that the arrangement of symbols within the given incomplete latin square can determine the embeddability. All known examples where the arrangement is a factor share the common feature that one symbol is prescribed to appear exactly once in the diagonal of $T$ outside of $L$. We show if the prescribed diagonal contains a symbol required to appear exactly once on the diagonal and $t ≤ 2n$, then there always exists a incomplete latin square satisfying the known numerical necessary conditions that is non-embeddable. Also, we solve a conjecture made over 30 years ago stating it is only this feature that prevents numerical conditions sufficing for all $t ≥ n$. Thus providing necessary and sufficient numerical conditions for the embedding of an incomplete latin square $L$ of order $n$ into a latin square $T$ of order $t$ for all $t ≥ n$ in which the diagonal of $T$ outside of $L$ is prescribed in the case where no symbol is required to appear exactly once in the diagonal of $T$ outside of $L$.

The presentation then briefly summarizes the second problem that concerns (not necessarily proper) $s$-edge-colorings of $K_{v}$ in which, for all  $u ∈ V(K_{v})$, the edges incident with $u$ are colored using exactly $p$ colors. In the spirit of proper edge-colorings, such $(s, p)$-edge-colorings are required to be equitable: the edges at each vertex are shared evenly among $p$ colors. Results and future directions are stated.