Extremal jumps of the Hall number

Abstract

Abstract 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 Hall number of G is the smallest positive integer k such that G admits a list coloring provided that | L ( v ) | ⩾ k for every vertex v and, for every X ⊆ V , the sum—over all colors c—of the maximum number of independent vertices in X whose lists contain c, is at least | X | . We prove that every graph of order n ⩾ 3 has Hall number at most n − 2 . Combining this upper bound with a construction, we deduce 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 , and that this estimate is tight for all n. This solves two problems raised by Hilton, Johnson and Wantland [Discrete Applied Math. 94 (1999), 227–245].