Coloring and Labeling Problems on Graphs by Daniel

This thesis studies both several extremal problems about coloring of graphs and a labeling problem on graphs. We consider colorings of graphs that are either embeddable in the plane or have low maximum degree. We consider three problems: coloring the vertices of a graph so that no adjacent vertices receive the same color, coloring the edges of a graph so that no adjacent edges receive the same color, and coloring the edges of a graph so that neither adjacent edges nor edges at distance one receive the same color. We use the model where colors on vertices must be chosen from assigned lists and consider the minimum size of lists needed to guarantee the existence of a proper coloring. More precisely, a list assignment function L assigns to each vertex a list of colors. A proper L-coloring is a proper coloring such that each vertex receives a color from its list. A graph is k-list-colorable if it has an L-coloring for every list assignment L that assigns each vertex a list of size k. The list chromatic number χ l (G) of a graph G is the minimum k such that G is k-list-colorable. We also call the list chromatic number the choice number of the graph. If a graph is k-list-colorable, we call it k-choosable. The elements of a graph are its vertices and edges. A proper total coloring of a graph is a coloring of the elements so that no adjacent elements and no incident elements receive the same color. The total list-chromatic number is the minimum list size that guarantees the existence of a proper total coloring. We give a linear-time algorithm to find a proper total coloring from lists of size 2∆(G) − 1. When ∆(G) = 4, our algorithm improves the best known upper bound. When ∆(G) ∈ {5, 6} our algorithm matches the best known upper bound and runs faster than the best previously known algorithm. The square of a graph G is the graph obtained from G by adding the edge xy whenever the distance between x and y in G is 2. We study the list chromatic numbers of squares of subcubic graphs; a graph is subcubic if it has maximum degree at most 3. We show that the square of every subcubic graph other than the Petersen graph is 8-list-colorable. For planar graphs with large girth, we use the discharging method …