5931910
The Hall Number, the Hall Index, and the Total Hall Number of a Graph
We discuss the relationship between the Hall number of a graph and other well-known graph parameters, including the choice number and the chromatic number. We also discuss the edge and total analogues of these relationships.
32175297
Which Graphs Have Hall Number Two?
33056920
Hall's condition for list-coloring, and the Hall parameters: recent developments
The study of list-colorings of simple graphs was originated by Erdős et al. [7] and independently by Vizing [25]. Erdős et al. had been inspired by the famous (and now solved [11]) problem of Je6rey Dinitz, posed at the Tenth Southeastern Conference on Combinatorics, Graph Theory, and Computing at Boca Raton in April, 1979: given an m × m array of m-sets, can representatives of these sets be chosen so that the representatives in each row and in each column are distinct? To see how Erdős, Rubin, and Taylor made this problem the progenitor of a small but busy combinatorial research industry, we will need some de=nitions. Throughout, G=(V; E) will be a simple graph, C will be some set of colors, or symbols, F(C) will be the collection of =nite subsets of C, and L :V → F(C) will be a function called a list assignment to the vertices of G. (Unless otherwise speci=ed, C will be in=nite.) A proper L-coloring of G is a function ’ :V → C such that (i) ’(v)∈L(v) for all v∈V and (ii) if uv∈E, then ’(u) =’(v). Suppose that f :V → P= [1; 2; : : : ] is a function. G is f-choosable if and only if there is a proper L-coloring of G whenever |L(v)|¿f(v) for all v∈V . The choice number of G (also called the list-chromatic number of G), denoted c(G) (sometimes ‘(G)), is the smallest constant c∈P such that G is c-choosable. Clearly (G)6 c(G), and it is shown in [7] that c(G) can be as much greater than (G) as desired. [In fact, in [20] it is shown, using a hard result of Erdős [6], that c(Km;m) ∼ log2m, while, of course, (Km;m)= 2. The problem of evaluating d(n)=max[c(G)= (G);G is simple, and |V (G)|= n] is open. We will, throughout, refer to the edge-analogue of a graph parameter “P number” de=ned with reference to the vertices of the graph, as the “P index” of the graph, and
40567017
Coloring the vertices of a graph with measurable sets in a probability space
We introduce a class of "chromatic" graph parameters that includes the chromatic number, the circular chromatic number, the fractional chromatic number, and an uncountable horde of others. We prove some basic results about this class and pose some problems.
19377592
Problems on Hall t-chromaticity
13138081
The structure of (t, r)-regular graphs of large order
A graph is (t,r)-regular iff it has at least one independent t-set of vertices and the open neighborhood of any such set contains exactly r vertices. Our goal is to show that when t>=3 and the order is sufficiently large, then the structure of (t,r)-regular graphs is similar to, but not exactly the same as the structure of (2,r)-regular graphs as derived by Faudree and Knisley. That is, there is an ''almost'' complete kernel of order at most r surrounded by satellite cliques, all of the same order, which are ''mostly'' joined to the kernel.