Four pages are necessary and sufficient for planar graphs

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. Given a list assignment L = {L(v)|v∈V} of G, we say G is acyclically L-list colorable if there exists a proper acyclic coloring π of G such that π(v)∈L(v) for ...

A graph G is k-degenerate if every subgraph of G has a vertex of degree at most k. It is known that every planar graph of girth 6 (equivalently, without 3-, 4-, and 5-cycles) is 2-degenerate. On the other hand, there exist planar ...

A (k, d)-list assignment L of a graph is a function that assigns to each vertex v a list L(v) of at least k colors satisfying $$|L(x)\cap L(y)|\le d$$|L(x)źL(y)|≤d for each edge xy. An L-coloring is a vertex coloring $$\pi $$ź such that $$\pi (v) \in L(...