Octahedralizing 3-Colorable 3-Polytopes

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, ...

Constructions of 3-Colorable Cores

Bicircular matroids are 3-colorable

Hugo Hadwiger proved that a graph that is not 3-colorable must have a K4minor and conjectured that a graph that is not k-colorable must have a Kk+1minor. By using the Hochstättler-Nešetřil definition for the chromatic number of an oriented matroid, we formulate a generalized version of Hadwiger’s conjecture that might hold for the class of oriented matroids. In particular, it is possible that e...

Acyclically 3-Colorable Planar Graphs

In this paper we study the planar graphs that admit an acyclic 3-coloring. We show that testing acyclic 3-colorability is NP-hard, even for planar graphs of maximum degree 4, and we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Further, we show that every planar graph has a subdivision with one vertex per edge that admits an acyclic 3-colori...

On uniquely 3-colorable planar graphs

A k-chromatic graph G is called uniquely k-colorable if every k-coloring of the vertex set V of G induces the same partition of V into k color classes. There is an innnite class C of uniquely 4-colorable planar graphs obtained from the K 4 by repeatedly inserting new vertices of degree 3 in triangular faces. In this paper we are concerned with the well-known conjecture (see 6]) that every uniqu...