Graphs and obstructions in four dimensions

Graphs and obstructions in four dimensions

For any graphG= (V ,E) without loops, let C2(G) denote the regular CW-complex obtained from G by attaching to each circuit C of G a disc. We show that if G is the suspension of a flat graph, then C2(G) has an embedding into 4-space. Furthermore, we show that for any graph G in the collection of graphs that can be obtained from K7 and K3,3,1,1 by a series of Y and Y -transformations, C2(G) canno...

Vertex Decomposable Graphs and Obstructions to Shellability

Inspired by several recent papers on the edge ideal of a graph G, we study the equivalent notion of the independence complex of G. Using the tool of vertex decomposability from geometric combinatorics, we show that 5-chordal graphs with no chordless 4-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, exte...

Tree-depth of Graphs: Characterisations and Obstructions

Four-Loop Cusp Anomalous Dimension From Obstructions

Freddy Cachazo, ∗ Marcus Spradlin, † and Anastasia Volovich ‡ Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2J 2W9, Canada Brown University, Providence, Rhode Island 02912, USA Abstract We introduce a method for extracting the cusp anomalous dimension at L loops from four-gluon amplitudes inN = 4 Yang-Mills without evaluating any integrals that depend on the kinematical invari...

Obstructions to partitions of chordal graphs

Matrix partition problems generalize graph colouring and homomorphism problems, and occur frequently in the study of perfect graphs. It is difficult to decide, even for a small matrix M , whether the M partition problem is polynomial time solvable or NP-complete (or possibly neither), and whether M -partitionable graphs can be characterized by a finite set of forbidden induced subgraphs (or per...