Alexander Clow

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20212026

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10 papers · 1 filter

math.CO2026

Greedily Constructing Small Quasi-Kernels

Alexander Clow

In a digraph $D$ ,a quasi-kernel is an independent set $Q$ such that for every vertex $u$ , there is a vertex $v \in Q$ satisfying $dist (v, u) \leq 2$ . In 1974 Chvátal and Lovás…

math.CO2025

Improved Bounds for the Ultimate Independence Ratio of Odd Wheels

Alexander Clow, Hitesh Kumar, Shivaramakrishna Pragada

The ultimate independence ratio of a graph $G$ is defined as $I (G) = lim_{k \to \infty} \frac{α ( G ^{□ k} )}{∣ V ( G ) ∣ ^{k}},$ where $α (G^{□ k})$ is the independence…

math.CO2025

A Note on Large Degenerate Induced Subgraphs in Sparse Graphs

Alexander Clow, Sean Kim, Ladislav Stacho

Given a graph $G$ and a non-negative integer $d$ let $α_{d} (G)$ be the order of a largest induced $d$ -degenerate subgraph of $G$ . We prove that for any pair of non-negative integers…

math.CO2025

Cops and Robbers on Graphs with Path Constraints

Alexander Clow, Erin Meger

In 2019, Sivaraman conjectured that every $P_{k}$ -free graph has cop number at most $k - 3$ . In the same year, Liu proved this conjecture for $(P_{k}, claw)$ -free graphs. Recently…

math.CO2025

Tight Bounds for Cycle-Edge Decompositions and Covers

Saieed Akbari, Jonny Aloni, Arash Beikmohammadi +1

An old conjecture of Erd{ő}s and Gallai states that every $n$ vertex graph can be decomposed, that is $E (G)$ can be partitioned, into $O (n)$ cycles and edges. The covering version…

math.CO2025

Cops and Robbers, Clique Covers, and Induced Cycles

Alexander Clow, Imed Zaguia

We consider the Cops and Robbers game played on finite simple graphs. In a graph $G$ , the number of cops required to capture a robber in the Cops and Robbers game is denoted by $c(…