Simplicial Complexes of Graphs

• Main Author: Jonsson, Jakob (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 2008.
• Series: Lecture Notes in Mathematics, 1928
• Subjects: Mathematics.; Algebra.; Ordered algebraic structures.; Algebraic topology.; Discrete mathematics.; Combinatorics.; Discrete Mathematics.; Algebraic Topology.; Order, Lattices, Ordered Algebraic Structures.

A graph complex is a finite family of graphs closed under deletion of edges. Graph complexes show up naturally in many different areas of mathematics, including commutative algebra, geometry, and knot theory. Identifying each graph with its edge set, one may view a graph complex as a simplicial comp...

Table of Contents:

• and Basic Concepts
• and Overview
• Abstract Graphs and Set Systems
• Simplicial Topology
• Tools
• Discrete Morse Theory
• Decision Trees
• Miscellaneous Results
• Overview of Graph Complexes
• Graph Properties
• Dihedral Graph Properties
• Digraph Properties
• Main Goals and Proof Techniques
• Vertex Degree
• Matchings
• Graphs of Bounded Degree
• Cycles and Crossings
• Forests and Matroids
• Bipartite Graphs
• Directed Variants of Forests and Bipartite Graphs
• Noncrossing Graphs
• Non-Hamiltonian Graphs
• Connectivity
• Disconnected Graphs
• Not 2-connected Graphs
• Not 3-connected Graphs and Beyond
• Dihedral Variants of k-connected Graphs
• Directed Variants of Connected Graphs
• Not 2-edge-connected Graphs
• Cliques and Stable Sets
• Graphs Avoiding k-matchings
• t-colorable Graphs
• Graphs and Hypergraphs with Bounded Covering Number
• Open Problems
• Open Problems.