Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Colored Saturation Parameters for Bipartite Graphs

Let F and H be fixed graphs and let G be a spanning subgraph of H. G is an F-free subgraph of H if F is not a subgraph of G. We say that G is an F-saturated subgraph of H if G is F-free and for any edge e in E(H)-E(G), F is a subgraph of G+e. The saturation number of F in K_{n,n}, denoted sat(K_{n,n}, F), is the minimum size of an F-saturated subgraph of K_{n,n}. A t-edge-coloring of a graph G is a labeling f: E(G) to [t], where [t] denotes the set { 1, 2, ..., t }. The labels assigned to the edges are called colors. A rainbow coloring is a coloring in which all edges have distinct colors. Given a family F of edge-colored graphs, a t-edge-colored graph H is (F, t)-saturated if H contains no member of F but the addition of any edge in any color completes a member of F. In this thesis we study the minimum size of ( F,t)-saturated subgraphs of edge-colored complete bipartite graphs. Specifically we provide bounds on the minimum size of these subgraphs for a variety of families of edge-colored bipartite graphs, including monochromatic matchings, rainbow matchings, and rainbow stars

Assessing the Computational Complexity of Multi-Layer Subgraph Detection

Multi-layer graphs consist of several graphs (layers) over the same vertex set. They are motivated by real-world problems where entities (vertices) are associated via multiple types of relationships (edges in different layers). We chart the border of computational (in)tractability for the class of subgraph detection problems on multi-layer graphs, including fundamental problems such as maximum matching, finding certain clique relaxations (motivated by community detection), or path problems. Mostly encountering hardness results, sometimes even for two or three layers, we can also spot some islands of tractability

Labeled Traveling Salesman Problems: Complexity and approximation

We consider labeled Traveling Salesman Problems, defined upon a complete graph of n vertices with colored edges. The objective is to find a tour of maximum or minimum number of colors. We derive results regarding hardness of approximation and analyze approximation algorithms, for both versions of the problem. For the maximization version we give a $1/2$-approximation algorithm based on local improvements and show that the problem is APX-hard. For the minimization version, we show that it is not approximable within $n^{1-\epsilon}$ for any fixed $\epsilon>0$. When every color appears in the graph at most $r$ times and $r$ is an increasing function of $n$, the problem is shown not to be approximable within factor $O(r^{1-\epsilon})$. For fixed constant $r$ we analyze a polynomial-time $(r +H_r)/2$ approximation algorithm, where $H_r$ is the $r$-th harmonic number, and prove APX-hardness for $r = 2$. For all of the analyzed algorithms we exhibit tightness of their analysis by provision of appropriate worst-case instances

Applications of Geometric and Spectral Methods in Graph Theory

Networks, or graphs, are useful for studying many things in today’s world. Graphs can be used to represent connections on social media, transportation networks, or even the internet. Because of this, it’s helpful to study graphs and learn what we can say about the structure of a given graph or what properties it might have. This dissertation focuses on the use of the probabilistic method and spectral graph theory to understand the geometric structure of graphs and find structures in graphs. We will also discuss graph curvature and how curvature lower bounds can be used to give us information about properties of graphs. A rainbow spanning tree in an edge-colored graph is a spanning tree in which each edge is a different color. Carraher, Hartke, and Horn showed that for n and C large enough, if G is an edge-colored copy of Kn in which each color class has size at most n/2, then G has at least [n/(C log n)] edge-disjoint rainbow spanning trees. Here we show that spectral graph theory can be used to prove that if G is any edge-colored graph with n vertices in which each color appears on at most δλ1/2 edges, where δ ≥ C log n for n and C sufficiently large and λ1 is the second-smallest eigenvalue of the normalized Laplacian matrix of G, then G contains at least [δλ1/ C log n] edge-disjoint rainbow spanning trees. We show how curvature lower bounds can be used in the context of understanding (personalized) PageRank, which was developed by Brin and Page. PageRank ranks the importance of webpages near a seed webpage, and we are interested in how this importance diffuses. We do this by using a notion of graph curvature introduced by Bauer, Horn, Lin, Lippner, Mangoubi, and Yau