Hall number for list colorings of graphs: Extremal results

Citations of this article
Mendeley users who have this article in their library.


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

Register to see more suggestions

Mendeley helps you to discover research relevant for your work.

Already have an account?

Save time finding and organizing research with Mendeley

Sign up for free