A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem states that every planar triangle-free graph is 3-colorable. However, Voigt [13] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangl...

Abstract Recently, Alon, Cambie, and Kang introduced asymmetric list coloring of bipartite graphs, where the size each vertex’s depends on its part. For complete we fix sizes one part consider resulting asymptotics, revealing an invariant quantity instrumental in determining choosability across most parameter space. By connecting this to a simple question independent sets hypergraphs, strengthe...

We extend a characterization of degree-choosable graphs due to Borodin [1], and Erdős, Rubin and Taylor [2], to circular list-colorings.

The choice number of a hypergraph H = (V, E) is the least integer s for which for every family of color lists S = {S(v) : v ∈ V }, satisfying |S(v)| = s for every v ∈ V , there exists a choice function f so that f(v) ∈ S(v) for every v ∈ V , and no edge of H is monochromatic under f . In this paper we consider the asymptotic behavior of the choice number of a random k-uniform hypergraph H(k, n,...

A proper vertex coloring of a non oriented graph G = (V, E) is linear if the graph induced by the vertices of two color classes is a forest of paths. A graph G is L-list colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is L-list colorable for every list assignment with |L(v)| ≥ k for all v ∈ V , then G...

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 vertexcoloring (with positive integers) of G such that c(v) ∈ S(v) whenever v ∈ V (G) and |c(u)− c(w)| 6∈ 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-li...

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′ = ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H )= (H ) for many “small” graphs H , including in9ations of all circuit...