A graph G = (V, E) is called (k, k′)-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers, there is a mapping f : V ∪ E → R such that f(y) ∈ L(y) for any y ∈ V ∪ E and for any two adjacent vertices x, x′, ∑ e∈E(x) f(e)+f(x) 6= ∑ e∈E(x′) f(e)+f(x ′). We ...

A p-list assignment L of a graph G assigns to each vertex v of G a set L(v) ⊆ {0, 1, . . . , p − 1} of permissible colors. We say G is L-(p, q)-colorable if G has a (p, q)-coloring h such that h(v) ∈ L(v) for each vertex v. The circular list chromatic number χc,l(G) of a graph G is the infimum of those real numbers t for which the following holds: For any p, q, for any p-list assignment L with ...

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each k ∈ {3, 4, 5, 6}, every signed planar graph without circuits of length k is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of le...

A graph G is (a, b)-choosable if for any assignment of a list of a colors to each of its vertices there is a subset of b colors of each list so that subsets corresponding to adjacent vertices are disjoint. It is shown that for every graph G, the minimum ratio a/b where a, b range over all pairs of integers for which G is (a, b)-choosable is equal to the fractional chromatic number of G.

A graph is k-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is k-choosable for k ≥ 3, and this problem is considered strictly harder than the k-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we ...

We show that every graph with two crossings is 5-choosable. We also prove that every graph which can be made planar by removing one edge is 5-choosable. Key-words: list colouring, choosability, crossing number † Universidade Federal do Ceará, Departamento de Computaçao, Bloco 910, Campus do Pici, Fortaleza, Ceará, CEP 60455-760, Brasil. [email protected]; Partially supported by CNPq/Brazil. ‡ P...

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