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Hall numbers of some complete $k-$partite graphs

Julian A. Allagan

TL;DR

The paper investigates Hall numbers for certain complete multipartite graphs and proves that for graphs of the form $G=K(m,2,\ldots,2)$ with $m\ge 2$, the Hall number equals the list-chromatic number, $h(G)=ch(G)$. It employs explicit list constructions and Hall's condition, together with known results on choice numbers of multipartite graphs, to establish exact values and derive corollaries for related graphs such as $K(3,2,\ldots,2)$, $K(3,3,2,\ldots,2)$, and $K(4,2,\ldots,2)$, including parity-based formulas. The work provides both concrete examples (e.g., a standard $L_0$ for $K(2,2)$) and general constructions that pin down $h(G)$ for broad families, culminating in a conjecture that $h(G)=ch(G)$ for all complete multipartite graphs with all parts of size greater than 1. These results have implications for understanding when the Hall condition suffices for colorability and for the broader study of the extremal equation $\chi(G)=ch(G)$ in graph coloring theory.

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.

Hall numbers of some complete $k-$partite graphs

TL;DR

The paper investigates Hall numbers for certain complete multipartite graphs and proves that for graphs of the form with , the Hall number equals the list-chromatic number, . It employs explicit list constructions and Hall's condition, together with known results on choice numbers of multipartite graphs, to establish exact values and derive corollaries for related graphs such as , , and , including parity-based formulas. The work provides both concrete examples (e.g., a standard for ) and general constructions that pin down for broad families, culminating in a conjecture that for all complete multipartite graphs with all parts of size greater than 1. These results have implications for understanding when the Hall condition suffices for colorability and for the broader study of the extremal equation in graph coloring theory.

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 , , are equal to their choice numbers.
Paper Structure (5 sections, 4 theorems, 6 equations, 1 figure)

This paper contains 5 sections, 4 theorems, 6 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Hall's Theorem
  3. Hall numbers of some complete multipartite graphs
  4. Example
  5. Some Hall numbers

Key Result

Theorem 1

(P. Hall Hall). Suppose $A_1,A_2,\ldots,A_n$ are (not necessarily distinct) finite sets. There exist distinct elements $a_1, a_2,\ldots,a_n$ such that $a_i\in A_i$, $i = 1, 2,\ldots, n$, if and only if for each $J \subseteq \{1, 2,\ldots , n\}$, $| \bigcup_ {j\in J} A_j | \ge |J|$.

Figures (1)

  • Figure 1: A list assignment to K(2,2).

Theorems & Definitions (10)

  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • Theorem 3
  • proof
  • proof
  • Theorem 4
  • proof
  • proof