List colorings with distinct list sizes, the case of complete bipartite graphs

General and acyclic sum-list-colouring of graphs

“…In colouring terms we allow that vertices of one colour class induce a graph that has some previously described properties, so it is not necessarily edgeless. Our investigation relates to the notions considered in the literature [2,7,9,11,12,14].…”

“…In [6] Erdös et al examined list colourings of complete bipartite balanced graphs to observe the difference between the concepts of proper colouring and list colouring. Next it was proved that the choice number (the minimum number d ∈ N such that for the constant function f ≡ d a graph is (f, O)-choosable) depends on an average degree of a graph [1] unlike the sum-choice-number (O-sum-choice-number) [7] and the last observation was based on the behaviour of the sum-choice-number of the unbalanced complete bipartite graphs. Some exact values of the sum-choicenumber of K p,q , but only for p ∈ {1, 2, 3} were given ( [2,9]).…”

List-coloring and sum-list-coloring problems on graphs

“…See [30] for more on this result. Alon [4] showed that χ l is bounded below by a function of the average degree: Theorem 1.16 (Alon [4]).…”

“…Füredi and Kantor [30] proved the following: 30]). There exist constants c 1 , c 2 such that for all m ≥ 4 and n ≥ 50m…”

“…It is a fairly new topic in graph theory, so there is much to be discovered. For more on sum-list-coloring see also [35,36,37,13,30]. In particular, [36] is a survey of all sum-list-coloring results up to 2007.…”