The Hall number, the Hall index, and the total Hall number of a graph

Hilton, A. J. W.; Johnson, Peter

doi:10.1016/s0166-218x(99)00023-2

Cited by 17 publications

(22 citation statements)

References 10 publications

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“…Thus the edge list colouring conjecture is that c 0 ðGÞ ¼ 0 ðGÞ, and Galvin's theorem is that this is true when G is a bipartite graph. In Section 2, we show (in addition to the items mentioned above) that it is possible for the Hall index to be reduced by an arbitrarily large amount upon the removal of an edge (the analogue for the Hall number under removal of a vertex was already known [15], but we also give a simpler proof of this fact).…”

Section: Definition and Known Resultsmentioning

confidence: 88%

“…s Ã ðGÞ, let s k ðGÞ be defined in the same way as sðGÞ was, except that the stock of colours is restricted so that it has cardinality k. The following theorem, proved in [14] and [20], provides a formula for sðGÞ and s k ðGÞ. This theorem was proved partly in [17] and partly in [15] for the unrestricted parameters, but the proof for the restricted parameters is nearly identical.…”

Section: Definition and Known Resultsmentioning

confidence: 97%

“…In an attempt to throw more light on the parameter cðGÞ, Hilton and Johnson introduced [13] the Hall number hðGÞ of G, and have studied it, with others, in [12][13][14][15][16][17][18][19]; see also [5,6,7,10,22]. If G is L-colourable then clearly X ð; L; HÞ !…”

Section: Definition and Known Resultsmentioning

confidence: 99%

“…Concerning the Hall number, it is shown in [15] that it is possible for hðGÞ À hðG À eÞ to be arbitrarily large, and, surprisingly, it is also possible for hðG À eÞ À hðGÞ to be arbitrarily large. The behaviour of the Hall number under removal of vertices is more constrained.…”

Section: A the Hall Index Upon Removal Of An Edgementioning

confidence: 99%

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Hall parameters of complete and complete bipartite graphs

Cropper

Hilton

2002

Journal of Graph Theory

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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

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Section: Definition and Known Resultsmentioning

confidence: 88%

Section: Definition and Known Resultsmentioning

confidence: 97%

Section: Definition and Known Resultsmentioning

confidence: 99%

Section: A the Hall Index Upon Removal Of An Edgementioning

confidence: 99%

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Hall parameters of complete and complete bipartite graphs

Cropper

Hilton

2002

Journal of Graph Theory

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“…Results on this family of graphs characterizing choosability in terms of given lists with a condition like Hall's Theorem are given in [4]. See also [3] and [5] for more references to related work.…”

Section: Introductionmentioning

confidence: 99%

Sum List Coloring Block Graphs

Isaak

2004

Graphs and Combinatorics

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Abstract. A graph is f -choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. We characterize f -choosable functions for block graphs (graphs in which each block is a clique, including trees and line graphs of trees).The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f . The sum choice number of any graph is at most the number of vertices plus the number of edges. We show that this bound is tight for block graphs.

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Characterization of graphs with hall number 2

Eslahchi

Johnson

2003

Journal of Graph Theory

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Hall's condition is a simple requirement that a graph G and list assignment L must satisfy if G is to have a proper L-colouring. The Hall number of G is the smallest integer m such that whenever the lists on the vertices each has size at least m and Hall's condition is satisfied a proper Lcolouring exists. Hilton and P.D. Johnson introduced the parameter and showed that a graph has Hall number 1 if and only if every block is a clique. In this paper we give a forbidden-induced-subgraph characterization of graphs with Hall number 2. ß

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The Hall number, the Hall index, and the total Hall number of a graph

Cited by 17 publications

References 10 publications

Hall parameters of complete and complete bipartite graphs

Hall parameters of complete and complete bipartite graphs

Sum List Coloring Block Graphs

Characterization of graphs with hall number 2

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