Density of 5/2-critical graphs

Chapter 52. Critical Pathways

Burgeoning concerns regarding patient safety, variable health care quality and increasing health care costs have led to the introduction of clinical management tools that have their origins outside of the traditional health care sector. Primary among these innovations has been the implementation of critical pathways, administrative models that streamline work and production processes. Critical ...

On the edge-density of 4-critical graphs

Gallai conjectured that every 4-critical graph on n vertices has at least 53n − 23 edges. We prove this conjecture for 4-critical graphs in which the subgraph induced by vertices of degree 3 is connected.

The edge-density of 4-critical planar graphs

Critical graphs without triangles: An optimum density construction

We construct dense, triangle-free, chromatic-critical graphs of chromatic number k for all k ≥ 4. For k ≥ 6 our constructions have > ( 1 4 −ε)n edges, which is asymptotically best possible by Turán’s theorem. We also demonstrate (nonconstructively) the existence of dense k-critical graphs avoiding all odd cycles of length ≤ l for any l and any k ≥ 4, again with a best possible density of > ( 1 ...

On the Critical Density for Percolation in Random Geometric Graphs

Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density λ c for d-dimensional Poisson random geometric graphs in continuum percolation theory. In two-dimensional space with the Euclidean norm, simulation studies show λ c ≈ 1.44, while the best theoretical bounds obtained thus far...