Louis Golowich — Page 2

arXiv2017

New Classes of Set-Sequential Trees

A graph is called set-sequential if its vertices can be labeled with distinct nonzero vectors in $\mathbb{F}_2^n$ such that when each edge is labeled with the sum$\pmod{2}$ of its vertices, every nonzero vector in $\mathbb{F}_2^n$ is the label for either a single vertex or a single edge. We resolve certain cases of a conjecture of Balister, Gyori, and Schelp in order to show many new classes of trees to be set-sequential. We show that all caterpillars $T$ of diameter $k$ such that $k \leq 18$ or $|V(T)| \geq 2^{k-1}$ are set-sequential, where $T$ has only odd-degree vertices and $|T| = 2^{n-1}$ for some positive integer $n$. We also present a new method of recursively constructing set-sequential trees.

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arXiv2017

Maximum Size of a Family of Pairwise Graph-Different Permutations

Two permutations of the vertices of a graph $G$ are called $G$-different if there exists an index $i$ such that $i$-th entry of the two permutations form an edge in $G$. We bound or determine the maximum size of a family of pairwise $G$-different permutations for various graphs $G$. We show that for all balanced bipartite graphs $G$ of order $n$ with minimum degree $n/2 - o(n)$, the maximum number of pairwise $G$-different permutations of the vertices of $G$ is $2^{(1-o(1))n}$. We also present examples of bipartite graphs $G$ with maximum degree $O(\log n)$ that have this property. We explore the problem of bounding the maximum size of a family of pairwise graph-different permutations when an unlimited number of disjoint vertices is added to a given graph. We determine this exact value for the graph of 2 disjoint edges, and present some asymptotic bounds relating to this value for graphs consisting of the union of $n/2$ disjoint edges.

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