Hall number for list colorings of graphs: Extremal results

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 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 K 2 ∪ ( n − 2 ) K 1 , P 4 ∪ ( n − 4 ) K 1 , and C 5 ∪ ( n − 5 ) K 1 . 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.