5 research outputs found
5-choosability of graphs with crossings far apart
We give a new proof of the fact that every planar graph is 5-choosable, and
use it to show that every graph drawn in the plane so that the distance between
every pair of crossings is at least 15 is 5-choosable. At the same time we may
allow some vertices to have lists of size four only, as long as they are far
apart and far from the crossings.Comment: 55 pages, 11 figures; minor revision according to the referee
suggestion
5-Choosability of Planar-plus-two-edge Graphs
We prove that graphs that can be made planar by deleting two edges are 5-choosable. To arrive at this, first we prove an extension of a theorem of Thomassen. Second, we prove an extension of a theorem Postle and Thomas. The difference between our extensions and the theorems of Thomassen and of Postle and Thomas is that we allow the graph to contain an inner 4-list vertex. We also use a colouring technique from two papers by Dvořák, Lidický and Škrekovski, and independently by Compos and Havet
Choosability of Graphs with Bounded Order: Ohba's Conjecture and Beyond
The \emph{choice number} of a graph , denoted , is the minimum
integer such that for any assignment of lists of size to the vertices
of , there is a proper colouring of such that every vertex is mapped to
a colour in its list. For general graphs, the choice number is not bounded
above by a function of the chromatic number.
In this thesis, we prove a conjecture of Ohba which asserts that
whenever . We also prove a
strengthening of Ohba's Conjecture which is best possible for graphs on at most
vertices, and pose several conjectures related to our work.Comment: Master's Thesis, McGill Universit
Graphs with two crossings are 5-choosable
A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. Thomassen’s theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with at most two crossings is 5-choosable