Critically paintable, choosable or colorable graphs

Planar Graphs of Girth at least Five are Square (∆ + 2)-Choosable

We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square (∆ + 2)-colorable for large enough ∆. In fact, we prove the stronger statement that such graphs are square (∆+2)-choosable and even square (∆+2)-paintable.

Characterization of $(2m, m)$-Paintable Graphs

In this paper, we prove that for any graph G and any positive integer m, G is (2m,m)-paintable if and only if G is 2-paintable. It was asked by Zhu in 2009 whether k-paintable graphs are (km,m)-paintable for any positive integer m. Our result answers this question in the affirmative for k = 2.

Coloring 3-Colorable Graphs

Graph coloring in general is an extremely easy-to-understand yet powerful tool. It has wide-ranging applications from register allocation to image segmentation. For such a simple problem, however, the question is surprisingly intractable. In this section I will introduce the problem formally, as well as present some general background on graph coloring. There are several ways to color a graph, ...

Planar graphs with square or cube root are four-colorable

Tiling Tripartite Graphs with 3-Colorable Graphs

For any positive real number γ and any positive integer h, there is N0 such that the following holds. Let N > N0 be such that N is divisible by h. If G is a tripartite graph with N vertices in each vertex class such that every vertex is adjacent to at least (2/3 + γ)N vertices in each of the other classes, then G can be tiled perfectly by copies of Kh,h,h. This extends the work in [Discrete Mat...