List Coloring Some Classes of 1-Planar Graphs

In list coloring we are given a graph G and a list assignment for G which assigns to each vertex of G a list of possible colors. We wish to find a coloring of the vertices of G such that each vertex uses a color from its list and adjacent vertices are given different colors. In this thesis we study the problem of list coloring 1-planar graphs, i.e., graphs that can be drawn in the plane such that any edge intersects at most one other edge. We also study the closely related problem of simultaneously list coloring the vertices and faces of a planar graph, known as coupled list coloring. We show that 1-planar bipartite graphs are list colorable whenever all lists are of size at least four, and further show that this coloring can be found in linear time. In pursuit of this result, we show that the previously known edge partition of a 1-planar graph into a planar graph and a forest can be found in linear time. A wheel graph consists of a cycle of vertices, all of which are adjacent to an additional center vertex. We show that wheel graphs are coupled list colorable when all lists are of size five or more and show that this coloring can be found in linear time. Possible extensions of this result to planar partial 3-trees are discussed. Finally, we discuss the complexity of list coloring 1-planar graphs, both in parameterized and unparameterized settings

Extremal problems involving forbidden subgraphs

In this thesis, we study extremal problems involving forbidden subgraphs. We are interested in extremal problems over a family of graphs or over a family of hypergraphs. In Chapter 2, we consider improper coloring of graphs without short cycles. We find how sparse an improperly critical graph can be when it has no short cycle. In particular, we find the exact threshold of density of triangle-free $(0,k)$-colorable graphs and we find the asymptotic threshold of density of $(j,k)$-colorable graphs of large girth when $k\geq 2j+2$. In Chapter 3, we consider other variations of graph coloring. We determine harmonious chromatic number of trees with large maximum degree and show upper bounds of $r$-dynamic chromatic number of graphs in terms of other parameters. In Chapter 4, we consider how dense a hypergraph can be when we forbid some subgraphs. In particular, we characterize hypergraphs with the maximum number of edges that contain no $r$-regular subgraphs. We also establish upper bounds for the number of edges in graphs and hypergraphs with no edge-disjoint equicovering subgraphs

Graph Powers: Hardness Results, Good Characterizations and Efficient Algorithms

Given a graph H = (V_H,E_H) and a positive integer k, the k-th power of H, written H^k, is the graph obtained from H by adding edges between any pair of vertices at distance at most k in H; formally, H^k = (V_H, {xy | 1 <= d_H (x, y) <= k}). A graph G is the k-th power of a graph H if G = H^k, and in this case, H is a k-th root of G. Our investigations deal with the computational complexity of recognizing k-th powers of general graphs as well as restricted graphs. This work provides new NP-completeness results, good characterizations and efficient algorithms for graph powers

Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS'09)

The Symposium on Theoretical Aspects of Computer Science (STACS) is held alternately in France and in Germany. The conference of February 26-28, 2009, held in Freiburg, is the 26th in this series. Previous meetings took place in Paris (1984), Saarbr¨ucken (1985), Orsay (1986), Passau (1987), Bordeaux (1988), Paderborn (1989), Rouen (1990), Hamburg (1991), Cachan (1992), W¨urzburg (1993), Caen (1994), M¨unchen (1995), Grenoble (1996), L¨ubeck (1997), Paris (1998), Trier (1999), Lille (2000), Dresden (2001), Antibes (2002), Berlin (2003), Montpellier (2004), Stuttgart (2005), Marseille (2006), Aachen (2007), and Bordeaux (2008). ..