1,199 research outputs found
Constructing graphs with no immersion of large complete graphs
In 1989, Lescure and Meyniel proved, for , that every -chromatic
graph contains an immersion of , and in 2003 Abu-Khzam and Langston
conjectured that this holds for all . In 2010, DeVos, Kawarabayashi, Mohar,
and Okamura proved this conjecture for . In each proof, the
-chromatic assumption was not fully utilized, as the proofs only use the
fact that a -critical graph has minimum degree at least . DeVos,
Dvo\v{r}\'ak, Fox, McDonald, Mohar, and Scheide show the stronger conjecture
that a graph with minimum degree has an immersion of fails for
and with a finite number of examples for each value of ,
and small chromatic number relative to , but it is shown that a minimum
degree of does guarantee an immersion of .
In this paper we show that the stronger conjecture is false for
and give infinite families of examples with minimum degree and chromatic
number or that do not contain an immersion of . Our examples
can be up to -edge-connected. We show, using Haj\'os' Construction, that
there is an infinite class of non--colorable graphs that contain an
immersion of . We conclude with some open questions, and the conjecture
that a graph with minimum degree and more than
vertices of degree at least has an immersion of
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
Realizing Health Reform's Potential: Small Businesses and the Affordable Care Act of 2010
Examines current coverage of workers in small firms, insurers' administrative costs, and healthcare reform provisions such as small-business tax credits to offset premiums and exemption from shared responsibility payments and the estimated impact of each
The 2008 Presidential Candidates' Health Reform Proposals: Choices for America
Looks at key differences in the 2008 presidential candidates' health insurance reforms and evaluates them against the Commonwealth Fund Commission on a High Performance Health System's principles. Estimates each plan's impact on the number of uninsured
Squeezed: Why Rising Exposure to Health Care Costs Threatens the Health and Financial Well-Being of American Families
Examines U.S. healthcare costs compared with other industrialized countries, individual health insurance coverage, individual market regulations, and the impact of high deductible plans on the health of individuals with chronic disease
Gaps in Health Insurance: An All-American Problem
Presents findings from a survey that examines health insurance coverage, rising healthcare costs, and the health and financial consequences to families that experience breaks in insurance
Health Coverage for Aging Baby Boomers: Findings From The Commonwealth Fund Survey of Older Adults
Examines healthcare quality and access by baby boomers in working families. Offers recommendations for expanding coverage, including options for savings accounts and early participation in Medicare
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