arXiv subjects

Isaiah Hollars

Publications and source records attributed to Isaiah Hollars.

4 recordsLinked to original sources

arXiv2026

Hitting all maximal independent sets in $c$-hollow graphs

Fix a constant $c$ with $0<c<1$. We say a graph $G$ on $n$ vertices is $c$-hollow if every maximal independent set of $G$ has size at least $cn$. Denote by $\tau(G)$ the size of a smallest set of vertices $T\subseteq V(G)$ such that every maximal independent set in $G$ intersects $T$, i.e., $T$ is a transversal for the family of maximal independent sets. In 1991, Bollob\'{a}s, Erd\H{o}s, and Tuza conjectured that if $G$ is $c$-hollow, then $\tau(G)=o(n)$. Using a random construction, we show there exist $c$-hollow graphs with $\tau(G)=\Omega\left(\frac{n^{1/3}}{\log n }\right)$, establishing the first nontrivial lower bound constraining the conjecture and complementing a closely related lower bound due to Alon for maximum independent sets. We also show the conjecture holds in a strong form for the class of cographs and split graphs.

math.CO↗

arXiv2026

Small $q$-kernels in digraphs with minimum in-degree $\delta$

For a digraph $D$, a subset $Q\subseteq V(D)$ is called a $q$-kernel if $Q$ is an independent set and all vertices in $V(D)$ are reachable from $Q$ via a directed path of length at most $q$. Given integers $q\geq 2$ and $\delta\geq 1$, Spiro arXiv:2404.07305 [math.CO] posed the question: what is the smallest constant $c_{\delta,q}$ such that every digraph $D$ with minimum in-degree $\delta$ has a $q$-kernel of size at most $c_{\delta,q}|V(D)|$? We show the constants $c_{\delta,q}$ are monotone in both $\delta$ and $q$, and we improve upon the known upper bounds for $c_{\delta,q}$. Our main results show $\frac{1}{\delta+1} \leq c_{\delta,q}\leq \frac{1}{\lfloor\sqrt{\delta+1}\rfloor+1}$ for all $q \geq 3$ and $\delta \geq 1$, and $ c_{\delta,q}=\frac{1}{\delta+1}$ whenever $\delta \geq 1$ and $q \geq \left\lceil\frac{3\delta}{2}\right\rceil + 1$.

math.CO↗

arXiv2025

Pancyclicity in hypergraphs with large uniformity

A Berge cycle of length $\ell$ in a hypergraph $\mathcal{H}$ is a sequence of alternating vertices and edges $v_0e_0v_1e_1...v_\ell e_\ell v_0$ such that $\{v_i,v_{i+1}\}\subseteq e_i$ for all $i$, with indices taken modulo $\ell$. For $n$ sufficiently large and $r\geq \lfloor\frac{n-1}{2}\rfloor-1$ we prove exact minimum degree conditions for an $n$-vertex, $r$-uniform hypergraph to contain Berge cycles of every length between $2$ and $n$. In conjunction with previous work, this provides sharp Dirac-type conditions for pancyclicity in $r$-uniform hypergraphs for all $3\leq r\leq n$ when $n$ is sufficiently large.

math.CO↗

arXiv2023

A natural bijection for contiguous pattern avoidance in words

Two words $p$ and $q$ are avoided by the same number of length-$n$ words, for all $n$, precisely when $p$ and $q$ have the same set of border lengths. Previous proofs of this theorem use generating functions but do not provide an explicit bijection. We give a bijective proof for all pairs $p, q$ that have the same set of proper borders, establishing a natural bijection from the set of words avoiding $p$ to the set of words avoiding $q$.

math.CO↗