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arXiv · 2509.01645

Formulas and Upper Bounds for the Carath{\'e}odory Number of Hamming Graphs

Ezequiel Dratman·Lucía M. González·Luciano N. Grippo
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Abstract

Let $G$ be a simple graph and let $S$ be a subset of its vertices. We say that $S$ is $P_3$-convex if every vertex $v \in V(G)$ that has at least two neighbors in $S$ also belongs to $S$. The $P_3$-hull set of $S$ is the smallest $P_3$-convex set of $G$ that contains $S$. Carath\'eodory number of a graph $G$, denoted by $c(G)$, is the smallest integer $c$ such that for every subset $S \subseteq V(G)$ and every vertex $p$ in the $P_3$-hull of $S$, there exists a subset $F \subseteq S$ with $|F| \leq c$ such that $p$ belongs to the $P_3$-hull of $F$. In this article, we present upper bounds and formulas for the $P_3$-Carath\'eodory number in Hamming graphs, which are defined as the Cartesian product of $n$ complete graphs.

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Ezequiel Dratman, Lucía M. González, Luciano N. Grippo. 2025-09-01. Formulas and Upper Bounds for the Carath{\'e}odory Number of Hamming Graphs. https://arxiv.org/abs/2509.01645

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