 
Summary: Tchoosability in graphs
Noga Alon 1
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact
Sciences, Tel Aviv University, Tel Aviv, Israel.
and
Ayal Zaks 2
Department of Statistics and Operations Research, Raymond and Beverly Sackler
Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel.
Abstract
Given a set of nonnegative integers T, and a function S which assigns a set of
integers S(v) to each vertex v of a graph G, an Slist Tcoloring c of G is a vertex
coloring (with positive integers) of G such that c(v) S(v) whenever v V (G)
and c(u)  c(w) T whenever (u, w) E(G). For a fixed T, the Tchoice number
Tch(G) of a graph G is the smallest number k such that G has an Slist Tcoloring
for every collection of sets S(v) of size k each. Exact values and bounds on the
Tr,schoice numbers where Tr,s = {0, s, 2s, . . . , rs} are presented for even cycles,
notably that Tr,sch(C2n) = 2r + 2 if n r + 1. More bounds are obtained by
applying algebraic and probabilistic techniques, such as that Tch(C2n) 2T if
0 T, and c1r log n Tr,sch(Kn,n) c2r log n for some absolute positive constants
c1, c2.
