Constructing graphs with no immersion of large complete graphs

In 1989, Lescure and Meyniel proved, for $d=5, 6$, that every $d$-chromatic graph contains an immersion of $K_d$, and in 2003 Abu-Khzam and Langston conjectured that this holds for all $d$. In 2010, DeVos, Kawarabayashi, Mohar, and Okamura proved this conjecture for $d = 7$. In each proof, the $d$-chromatic assumption was not fully utilized, as the proofs only use the fact that a $d$-critical graph has minimum degree at least $d - 1$. DeVos, Dvo\v{r}\'ak, Fox, McDonald, Mohar, and Scheide show the stronger conjecture that a graph with minimum degree $d-1$ has an immersion of $K_d$ fails for $d=10$ and $d\geq 12$ with a finite number of examples for each value of $d$, and small chromatic number relative to $d$, but it is shown that a minimum degree of $200d$ does guarantee an immersion of $K_d$. In this paper we show that the stronger conjecture is false for $d=8,9,11$ and give infinite families of examples with minimum degree $d-1$ and chromatic number $d-3$ or $d-2$ that do not contain an immersion of $K_d$. Our examples can be up to $(d-2)$-edge-connected. We show, using Haj\'os' Construction, that there is an infinite class of non-$(d-1)$-colorable graphs that contain an immersion of $K_d$. We conclude with some open questions, and the conjecture that a graph $G$ with minimum degree $d - 1$ and more than $\frac{|V(G)|}{1+m(d+1)}$ vertices of degree at least $md$ has an immersion of $K_d$

Progressive or Regressive? A Second Look at the Tax Exemption for Employer Sponsored Health Insurance Premiums

Examines the argument for capping the exemption of health insurance benefits from income tax and its potential effects on those already at risk of losing their coverage. Estimates the regressive impact by firm size, location, risk group, and income level

Health Care Opinion Leaders' Views on Health Spending and Reform Implementation

Presents survey responses from healthcare experts about support for the reform law's coverage expansion provisions, payment and delivery system reforms, the budget reduction framework that relies in part on Medicare and Medicaid savings, and other issues

Health Care Opinion Leaders' Views on Priorities for the Obama Administration

Presents results of a survey of experts on priorities in healthcare reform and related economic stimulus for the Barack Obama administration, including new regulations, employer financing of coverage, and an insurance exchange with a public option

Health Care Opinion Leaders' Views on Health Reform

Presents results of a survey of healthcare experts on insurance, payment, and delivery system reforms, including a national insurance exchange with broad authority, a public plan option with a bundled payment system, and a standard benefits package

On planar and dismantlable lattices

AbstractIt is shown that any finite, rank-connected, dismantlable lattice is lexicographically shellable (hence Cohen–Macaulay). A ranked, interval-connected lattice is shown to be rank-connected, but a rank-connected lattice need not be interval-connected. An example of a planar, rank-connected lattice that is not admissible is given

Wages, Health Benefits, and Workers' Health

Examines the divide in the U.S. labor market between higher wage earners with health insurance, and lower wage earners who often lack coverage and access to essential health care

A qualitative exploration of the impact of personal development in counselling training on the student counsellor's significant relationships: Should counsellor training come with a stronger warning or more support?

A small scale qualitative research study set out to explore the impact of the personal development element of counsellor training on the student counsellors’ significant relationship. Six qualified counsellors shared their own experiences of training, and the impact it had on their relationships, in a semi-structured, one to one interview. The data gathered was subjected to a form of grounded theory. The study concluded that the personal development element does have an impact on students’ relationships; some survived and others ended. Whilst this was generally perceived by the participants as positive, the study found a number of factors, resulting from personal development in counselling training, which did contribute to various stresses being placed on the participants’ relationships