Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids

This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory

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

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs

Two distinct crossings are independent if the end-vertices of the crossed pair of edges are mutually different. If a graph G has a drawing in the plane such that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. A proper total-k-coloring of a graph G is a mapping c : V (G) ∪ E(G) → {1, 2, . . . , k} such that any two adjacent elements in V (G) ∪ E(G) receive different colors. Let Σc(v) denote the sum of the color of a vertex v and the colors of all incident edges of v. A total-k-neighbor sum distinguishing-coloring of G is a total-k-coloring of G such that for each edge uv ∈ E(G), Σc(u) ≠ Σc(v). The least number k needed for such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χΣ″(G)\chi _\Sigma ^{''} ( G ) . In this paper, it is proved that if G is an IC-planar graph with maximum degree Δ(G), then chΣ″(G)≤max{Δ(G)+3, 17}ch_\Sigma ^{''} ( G ) \le \max \left\{ {\Delta ( G ) + 3,\;17} \right\} , where chΣ″(G)ch_\Sigma ^{''} ( G ) is the neighbor sum distinguishing total choosability of G

Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs

Two distinct crossings are independent if the end-vertices of the crossed pair of edges are mutually different. If a graph $G$ has a drawing in the plane such that every two crossings are independent, then we call $G$ a plane graph with independent crossings or IC-planar graph for short. A proper total-$k$-coloring of a graph $G$ is a mapping $c : V (G) \cup E(G) \rightarrow \{ 1, 2, . . ., k \}$ such that any two adjacent elements in $V (G) \cup E(G)$ receive different colors. Let $\Sigma_c (v)$ denote the sum of the color of a vertex $v$ and the colors of all incident edges of $v$. A total-$k$-neighbor sum distinguishing-coloring of $G$ is a total-$k$-coloring of $G$ such that for each edge $uv \in E(G)$, $\Sigma_c (u) \ne \Sigma_c (v)$. The least number $k$ needed for such a coloring of $G$ is the neighbor sum distinguishing total chromatic number, denoted by $\chi_\Sigma^{''} (G)$. In this paper, it is proved that if $G$ is an IC-planar graph with maximum degree $\Delta (G)$, then $ch_\Sigma^{''} (G) \le \text{max} \{ \Delta (G)+3, 17 \}$, where $ch_\Sigma^{''} (G)$ is the neighbor sum distinguishing total choosability of $G$

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). ..