On the penrose number of cubic diagrams
Discrete Mathematics - Graph colouring and variations
A solution to a colouring problem of P. Erdős
Discrete Mathematics - Special volume (part two) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs” (“The theory of regular graphs”)
Discrete Applied Mathematics - ARIDAM IV and V
Restricted colorings of graphs
Surveys in combinatorics, 1993
The list chromatic index of a bipartite multigraph
Journal of Combinatorial Theory Series B
Asymptotically good list-colorings
Journal of Combinatorial Theory Series A
Discrete Applied Mathematics
Kr-free uniquely vertex colorable graphs with minimum possible edges
Journal of Combinatorial Theory Series B
Combinatorics, Probability and Computing
A relation between choosability and uniquely list colorability
Journal of Combinatorial Theory Series B
Anti-magic graphs via the Combinatorial NullStellenSatz
Journal of Graph Theory
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In a seminal paper (Alon and Tarsi, 1992 [6]), Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of G obtained via their method, was later coined the Alon-Tarsi number of G and was denoted by AT(G) (see e.g. Jensen and Toft (1995) [20]). They have provided a combinatorial interpretation of this parameter in terms of the eulerian subdigraphs of an appropriate orientation of G. Their characterization can be restated as follows. Let D be an orientation of G. Assign a weight @w"D(H) to every subdigraph H of D: if H@?D is eulerian, then @w"D(H)=(-1)^e^(^H^), otherwise @w"D(H)=0. Alon and Tarsi proved that AT(G)=0. Shortly afterwards (Alon, 1993 [3]), for the special case of line graphs of d-regular d-edge-colorable graphs, Alon gave another interpretation of AT(G), this time in terms of the signed d-colorings of the line graph. In this paper we generalize both results. The first characterization is generalized by showing that there is an infinite family of weight functions (which includes the one considered by Alon and Tarsi), each of which can be used to characterize AT(G). The second characterization is generalized to all graphs (in fact the result is even more general-in particular it applies to hypergraphs). We then use the second generalization to prove that @g(G)=ch(G)=AT(G) holds for certain families of graphs G. Some of these results generalize certain known choosability results.