Hall numbers of some complete $k-$partite graphs

TL;DR

This paper investigates the Hall numbers of specific complete multipartite graphs, demonstrating that for graphs of the form K(m,2,...,2), the Hall number equals the choice number, thus linking these two graph parameters.

Contribution

It establishes the equality of Hall numbers and choice numbers for a class of complete multipartite graphs, providing new insights into their combinatorial properties.

Findings

Hall numbers of K(m,2,...,2) equal their choice numbers

Provides exact values for Hall numbers in these graphs

Links between Hall numbers and choice numbers are clarified

Abstract

The Hall number is a graph parameter closely related to the choice number. Here it is shown that the Hall numbers of the complete multipartite graphs $K(m,2,\ldots,2)$, $m\ge 2$, are equal to their choice numbers.