Hall parameters of complete and complete bipartite graphs

Abstract

Given a graph G, for each υ ∈V(G) let L(υ) be a list assignment to G. The well‐known choice number c(G) is the least integer j such that if |L(υ)| ≥j for all υ ∈V(G), then G has a proper vertex colouring ϕ with ϕ(υ) ∈ L (υ) (∀υ ∈V(G)). The Hall number h(G) is like the choice number, except that an extra non‐triviality condition, called Hall's condition, has to be satisfied by the list assignment. The edge‐analogue of the Hall number is called the Hall index, h′(G), and the total analogue is called the total Hall number, h″(G), of G. If the stock of colours from which L(υ) is selected is restricted to a set of size k, then the analogous numbers are called k‐restricted, or restricted, Hall parameters, and are denoted by hk(G), h′k(G) and h″k(G).Our main object in this article is to determine, or closely bound, h′(K), h″(Kn), h′(Km,n) and h′k(Km,n). We also answer some hitherto unresolved questions about Hall parameters. We show in particular that there are examples of graphs G with h′(G)−h′(G − e)>1. We show that there are examples of graphs G and induced subgraphs H with hk(G)<hk(H) [this phenomenon cannot occur with unrestricted Hall numbers]. We also give an example of a graph G and an integer k such that hk(G)<χ(G)<h(G). © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 208–237, 2002

An analogue of Ryser's Theorem for partial Sudoku squares

“…Hall's Condition for graphs has been the object of a certain amount of study (see e.g. [7,8,9,10,13,20,22,24,23]) and recently Hall's Condition for partial latin squares has been studied (see [5,19,24,23]). In every case that has been looked at so far, whenever there is a theorem saying that a certain kind of partial latin square can be completed if and only if such and such a condition is satisfied, then the condition (whatever it might be) can be replaced by Hall's Condition.…”

“…(This is Hall's Theorem [16], and an L-colouring in this case is a system of distinct representatives.) Hall's Condition for graphs has been the subject of several recent papers (see [7,8,9,10,13,19,20,21,22,24,23]). In general, Hall's Condition for graphs is not sufficient to ensure there is a list-colouring.…”

Hall's theorem and extending partial latinized rectangles

“…The utility of extensions of Hall's Condition is well-known, but the paper [10] was the first which actively engaged the attention of the second author and Johnson. Since then, they -and others -have explored various aspects, as have various papers more to do with graph theory, of which [3][4][5]13] are examples.…”