Given a graph G = (V, E) and sets L (v) of allowed colors for each v ∈ V, a list coloring of G is an assignment of colors φ (v) to the vertices, such that φ (v) ∈ L (v) for all v ∈ V and φ (u) ≠ φ (v) for all u v ∈ E. The choice number of G is the smallest natural number k admitting a list coloring for G whenever | L (v) | ≥ k holds for every vertex v. This concept has an interesting variant, called Hall number, where an obvious necessary condition for colorability is put as a restriction on the lists L (v). (On complete graphs, this condition is equivalent to the well-known one in Hall's Marriage Theorem.) We prove that vertex deletion or edge insertion in a graph of order n > 3 may make the Hall number decrease by as much as n - 3. This estimate is tight for all n. Tightness is deduced from the upper bound that every graph of order n has Hall number at most n - 2. We also characterize the cases of equality; for n ≥ 6 these are precisely the graphs whose complements are K2 ∪ (n - 2) K1, P4 ∪ (n - 4) K1, and C5 ∪ (n - 5) K1. Our results completely solve a problem raised by Hilton, Johnson and Wantland [A.J.W. Hilton, P.D. Johnson, Jr., E. B. Wantland, The Hall number of a simple graph, Congr. Numer. 121 (1996), 161-182, Problem 7] in terms of the number of vertices, and strongly improve some estimates due to Hilton and Johnson [A.J.W. Hilton, P.D. Johnson, Jr., The Hall number, the Hall index, and the total Hall number of a graph, Discrete Appl. Math. 94 (1999), 227-245] as a function of maximum degree. © 2009 Elsevier B.V. All rights reserved.
Tuza, Z. (2010). Hall number for list colorings of graphs: Extremal results. Discrete Mathematics, 310(3), 461–470. https://doi.org/10.1016/j.disc.2009.03.025