2002
DOI: 10.1002/jgt.10063
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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 Ha… Show more

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Cited by 2 publications
(3 citation statements)
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“…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.…”
Section: Analogues For Sudoku Squares Of Results About Partial Latin ...mentioning
confidence: 99%
See 1 more Smart Citation
“…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.…”
Section: Analogues For Sudoku Squares Of Results About Partial Latin ...mentioning
confidence: 99%
“…(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.…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%