Choosability of graphs with infinite sets of forbidden differences

Choosability of graphs with infinite sets of forbidden differences

On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A (k, d)-list assignment L of a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k, d)-choosable if there exists an L-coloring of G for every (k, d)-list assignment L. This...

(4, 2)-Choosability of Planar Graphs with Forbidden Structures

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any l∈{3,4,5,6,7}" role="presentation" style="box-sizing: border-box; display: inline; line-height: normal; ...

k-forested choosability of graphs with bounded maximum average degree

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prov...

Well-quasi-ordering Infinite Graphs with Forbidden Finite Planar Minor

We prove that given any sequence G\, Gi,... of graphs, where G\ is finite planar and all other G, are possibly infinite, there are indices ;', j such that i < j and G¡ is isomorphic to a minor of Gj . This generalizes results of Robertson and Seymour to infinite graphs. The restriction on G\ cannot be omitted by our earlier result. The proof is complex and makes use of an excluded minor theorem...