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T-choosability in graphs Noga Alon 1
 

Summary: T-choosability 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 S-list T-coloring 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 T-choice number
T-ch(G) of a graph G is the smallest number k such that G has an S-list T-coloring
for every collection of sets S(v) of size k each. Exact values and bounds on the
Tr,s-choice numbers where Tr,s = {0, s, 2s, . . . , rs} are presented for even cycles,
notably that Tr,s-ch(C2n) = 2r + 2 if n r + 1. More bounds are obtained by
applying algebraic and probabilistic techniques, such as that T-ch(C2n) 2|T| if
0 T, and c1r log n Tr,s-ch(Kn,n) c2r log n for some absolute positive constants
c1, c2.

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics