arXiv subjects
Publications and source records attributed to Chase Wilson.
Let $G$ be a graph and $w_k(G)$ denote the number of walks in $G$ of length $k$. For sequences $a_1, \cdots, a_n$ and $b_1, \cdots, b_n$ of non-negative integers such that $a_1 + \cdots + a_n = b_1 + \cdots + b_n$, we determine a simple necessary and sufficient condition on $a_1, \cdots, a_n, b_1, \cdots, b_n$ for the inequality \[ w_{a_1}(G) \cdots w_{a_n}(G) \geq w_{b_1}(G) \cdots w_{b_n}(G) \] to hold for any graph $G$.
An old problem in combinatorial group theory asks, given a group $\Gamma$ and a subset $S \subseteq \Gamma$, when does there exist an ordering $s_1, \cdots, s_k$ of the elements of $S$ such that the partial products $\prod_{i = 1}^j s_i$, $1 \leq j \leq k$, are all distinct. If such an ordering exists, we call $S$ rearrangeable. There have been many conjectures about rearrangeable subsets, the most general being that for every group, every subset not containing the identity element is rearrangeable. We prove an asymptotic version of this: For any Group $\Gamma$ and any subset $S \subseteq \Gamma$, there exists a rearrangeable set $S' \subseteq S$ such that $|S'| = |S| - o(|S|)$. To do this we build upon the work of Buci\'c, Frederickson, M\"uyesser, Pokrovskiy, and Yepremyan focusing on the following problem of independent interest in Graph Theory. If $G$ is a d-regular properly colored directed graph does there exist a rainbow path of length $d - 1$? We establish an asymptotic version of this, proving that $G$ contains a rainbow path of length $d - o(d)$. This solves two problems given by Buci\'c, Frederickson, et al. and proves the above result on rearrangeable subsets of groups by considering the Cayley graph of $\Gamma$ with (not necessarily generating) set $S$.
Mubayi and Verstraete conjectured that if $T$ is a tree on $t + 1$ vertices, then any $n$-vertex graph $G$ with average degree $d$ contains at least \[ n d(d - 1) \cdots (d - t + 1) \] labeled copies of $T$ as long as $d$ is sufficiently large compared to $t$. We prove this is true and show that when the diameter of $T$ is at least $3$, equality holds iff $G$ is the disjoint union of cliques of size $d + 1$. When the diameter is $2$, equality holds iff $G$ is $d$-regular.
A theorem of Shearer states that every $n$-vertex triangle-free graph of maximum degree $d \geq 2$ contains an independent set of size at least $(d\log d - d + 1)/(d - 1)^2 \cdot n$. Ajtai, Koml\'{o}s, Pintz, Spencer and Szemer\'{e}di proved that every $(r + 1)$-uniform $n$-vertex ``uncrowded'' hypergraph of maximum degree $d \geq 1$ has an independent set of size at least $c_r(\log d)^{1/r}/d^{1/r} \cdot n$ for some $c_r > 0$ depending only on $r$. Shearer asked whether his method for triangle-free graphs could be extended to uniform hypergraphs. In this paper, we answer this in the affirmative, thereby giving a short proof of the theorem of Ajtai, Koml\'{o}s, Pintz, Spencer and Szemer\'{e}di for a wider class of ``locally sparse'' hypergraphs.