Vizing [15] and Erdös et al. [8] independently introduce the idea of considering list-colouring and k-choosability. In the both papers the choosability version of Brooks’ theorem [4] was proved but the choosability version of Gallai’s theorem [9] was proved independently by Thomassen [14] and by Kostochka et al. [11]. In [3] some extensions of these two basic theorems to (P, k)-choosability hav...

Given a multigraph, suppose that each vertex is given a local assignment of k colours to its incident edges. We are interested in whether there is a choice of one local colour per vertex such that no edge has both of its local colours chosen. The least k for which this is always possible given any set of local assignments we call the conflict choosability of the graph. This parameter is closely...

The circular consecutive choosability chcc(G) of a graph G has been recently introduced in [2]. In this paper we prove upper bounds on chcc for series-parallel graphs, planar graphs and k-choosable graphs. Our bounds are tight for classes of series-parallel graphs and k-choosable graphs for k ≥ 3. Then we study the circular consecutive choosability of generalized theta graphs. Lower bounds for ...

We introduce and study adapted list coloring of graphs and hypergraphs. This is a generalization of ordinary list coloring and adapted coloring, and has more applications than these. We prove that the upper bounds on the adaptable choosability of graphs and uniform hypergraphs in terms of maximum degree are sufficiently stronger than those on the ordinary choosability, while the bounds in terms...

A graph G is (a, b)-choosable if for any color list of size a associated with each vertices, one can choose a subset of b colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence between the (a, b)-choosability of a graph and the (a, b)-choosability of one of its subgraphs called the extended core. As an application, this result allows to prove th...

It is proved that a planar graph G without five cycles is three degenerate, hence, four choosable, and it is also edge-(A( G) + l)h c oosable. @ 2002 Elsevier Science Ltd. All rights reserved. Keywords-Choosability, Edge choosability, Degeneracy, Planar graph.

We consider improper colorings (sometimes called generalized, defective or relaxed colorings) in which every color class has a bounded degree. We propose a natural extension of improper colorings: acyclic improper choosability. We prove that subcubic graphs are acyclically (3,1)∗-choosable (i.e. they are acyclically 3-choosable with color classes of maximum degree one). Using a linear time algo...

A multicircuit is a multigraph whose underlying simple graph is a circuit (a connected 2-regular graph). In this paper, the method of Alon and Tarsi is used to prove that all multicircuits of even order, and some regular and near-regular multicircuits of odd order have total choosability (i.e., list total chromatic number) equal to their ordinary total chromatic number. This completes the proof...