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Daniel McGinnis

Publications and source records attributed to Daniel McGinnis.

23 records · Page 2Linked to original sources
arXiv2021

Line transversals in families of connected sets the plane

We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that, under the same condition, there are four lines intersecting all the sets. In fact, we prove a colorful version of this result, under weakened conditions on the sets. A triple of sets $A,B,C$ in the plane is said to be a {\em tight} if $\textrm{conv}(A\cup B)\cap \textrm{conv}(A\cup C)\cap \textrm{conv}(B\cap C)\neq \emptyset.$ This notion was first introduced by Holmsen, where he showed that if $\mathcal{F}$ is a family of compact convex sets in the plane in which every three sets form a tight triple, then there is a line intersecting at least $\frac{1}{8}|\mathcal{F}|$ members of $\mathcal{F}$. Here we prove that if $\mathcal{F}_1,\dots,\mathcal{F}_6$ are families of compact connected sets in the plane such that every three sets, chosen from three distinct families $\mathcal{F}_i$, form a tight triple, then there exists $1\le j\le 6$ and three lines intersecting every member of $\mathcal{F}_j$. In particular, this improves $\frac{1}{8}$ to $\frac{1}{3}$ in Holmsen's result.

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arXiv2019

Extremal Problems Related to the Cardinality Redundance of Graphs

A dominating set of a graph $G$ is a set of vertices $D$ such that for all $v \in V(G)$, either $v \in D$ or $(v,d) \in E(G)$ for some $d \in D$. The cardinality redundance of a vertex set $S$, $CR(S)$, is the number of vertices in $V(G)$ such that $|N[x] \cap S| \geq 2$. The cardinality redundance of $G$ is the minimum of $CR(S)$ taken over all dominating sets $S$. A set that achieves $CR(G)$ is a $γ_{cr}$-set, and the size of the minimum $γ_{cr}$-set is $γ_{cr}(G)$. We give the maximum number of edges in a graph with a given number of vertices and given cardinality redundance. In the cases that $CR(G)=0$, $1$, or $2$, we give the minimum and maximum number of edges of graphs where $γ_{cr}(G)$ is fixed. We give the minimum and maximum values of $γ_{cr}(G)$ when the number of edges are fixed and $CR(G)=0,1$, and we give the maximum values of $γ_{cr}(G)$ when the number of edges are fixed and $CR(G)=2$.

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arXiv2018

A Method to Construct $1$-Rotational Factorizations of Complete Graphs and Solutions to the Oberwolfach Problem

The concept of a $1$-rotational factorization of a complete graph under a finite group $G$ was studied in detail by Buratti and Rinaldi. They found that if $G$ admits a $1$-rotational $2$-factorization, then the involutions of $G$ are pairwise conjugate. We extend their result by showing that if a finite group $G$ admits a $1$-rotational $k=2^nm$-factorization where $n\geq 1$, and $m$ is odd, then $G$ has at most $m(2^n-1)$ conjugacy classes containing involutions. Also, we show that if $G$ has exactly $m(2^n-1)$ conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a $1$-rotational $2n$-factorization under $G \times \mathbb{Z}_n$ given a $1$-rotational $2$-factorization under a finite group $G$. This construction, given a $1$-rotational solution to the Oberwolfach problem $OP(a_{\infty},a_1, a_2 \cdots, a_n)$, allows us to find a solution to $OP(2a_{\infty}-1,^2a_1, ^2a_2\cdots, ^2a_n)$ when the $a_i$'s are even ($i \neq \infty$), and $OP(p(a_{\infty}-1)+1, ^pa_1, ^pa_2 \cdots, ^pa_n)$ when $p$ is an odd prime, with no restrictions on the $a_i$'s.

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arXiv2018

On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points

A permutation graph $G_π$ is a simple graph with vertices corresponding to the elements of $π$ and an edge between $i$ and $j$ when $i$ and $j$ are inverted in $π$. A set of vertices $D$ is said to dominate a graph $G$ when every vertex in $G$ is either an element of $D$, or adjacent to an element of $D$. The domination number $γ(G)$ is defined as the cardinality of a minimum dominating set of $G$. A strong fixed point of a permutation $π$ of order $n$ is an element $k$ such that $π^{-1}(j)<π^{-1}(k)$ for all $j π^{-1}(k)$ for all $i>k$. In this article, we count the number of connected permutation graphs on $n$ vertices with domination number $1$ and domination number $\frac{n}{2}$. We further show that for a natural number $k\leq \frac{n}{2}$, there exists a connected permutation graph on $n$ vertices with domination number $k$. We find a closed expression for the number of permutation graphs dominated by a set with two elements, and we find a closed expression for the number of permutation graphs efficiently dominated by any set of vertices. We conclude by providing an application of these results to strong fixed points, proving some conjectures posed on the OEIS.

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