Coloring problems in graph theory

Abstract

<p>We consider two branches of coloring problems for graphs: list coloring and packing coloring. We introduce a new variation to list coloring which we call choosability with union separation: For a graph G, a list assignment L to the vertices of G is a (k,k+t)-list assignment if every vertex is assigned a list of size at least k and the union of the lists of each pair of adjacent vertices is at least k+t. We explore this new variation and offer comparative results to choosability with intersection separation, a variation that has been studied previously.</p> <p>Regarding packing colorings, we consider infinite lattice graphs and provide bounds to their packing chromatic numbers. We also provide algorithms for coloring these graphs. The lattices we color include two-layer hexagonal lattices as well as the truncated square lattice, a 3-regular lattice whose faces have length 4 and 8.</p>