scholarly journals A maximal zero-free interval for chromatic polynomials of bipartite planar graphs

2008 ◽  
Vol 308 (11) ◽  
pp. 2285-2287 ◽  
Author(s):  
F.M. Dong ◽  
K.M. Koh
Keyword(s):  
Planar Graphs ◽  
Chromatic Polynomials ◽  
Free Interval

scholarly journals A zero-free interval for chromatic polynomials

1992 ◽  
Vol 101 (1-3) ◽  
pp. 333-341 ◽  
Author(s):  
D.R. Woodall
Keyword(s):  
Chromatic Polynomials ◽  
Free Interval

scholarly journals Density of Chromatic Roots in Minor-Closed Graph Families

2018 ◽  
Vol 27 (6) ◽  
pp. 988-998 ◽  
Author(s):  
THOMAS J. PERRETT ◽  
CARSTEN THOMASSEN
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Classical Result ◽  
Golden Ratio ◽  
Chromatic Polynomial ◽  
Closed Graph ◽  
Small Interval ◽  
Chromatic Polynomials ◽  
Graph Families ◽  
Chromatic Roots

We prove that the roots of the chromatic polynomials of planar graphs are dense in the interval between 32/27 and 4, except possibly in a small interval around τ + 2 where τ is the golden ratio. This interval arises due to a classical result of Tutte, which states that the chromatic polynomial of every planar graph takes a positive value at τ + 2. Our results lead us to conjecture that τ + 2 is the only such number less than 4.

A Contribution to the Theory of Chromatic Polynomials

1954 ◽  
Vol 6 ◽  
pp. 80-91 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Connected Graph ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Chromatic Polynomials ◽  
Colour Problem ◽  
General Graphs

SummaryTwo polynomials θ(G, n) and ϕ(G, n) connected with the colourings of a graph G or of associated maps are discussed. A result believed to be new is proved for the lesser-known polynomial ϕ(G, n). Attention is called to some unsolved problems concerning ϕ(G, n) which are natural generalizations of the Four Colour Problem from planar graphs to general graphs. A polynomial χ(G, x, y) in two variables x and y, which can be regarded as generalizing both θ(G, n) and ϕ(G, n) is studied. For a connected graph χ(G, x, y) is defined in terms of the “spanning” trees of G (which include every vertex) and in terms of a fixed enumeration of the edges.

A Zero-Free Interval for Chromatic Polynomials of Graphs

1993 ◽  
Vol 2 (3) ◽  
pp. 325-336 ◽  
Author(s):  
Bill Jackson
Keyword(s):  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Free Interval

LetGbe a graph andP(G, t) be the chromatic polynomial ofG. It is known thatP(G, t) has no zeros in the intervals (−∞, 0) and (0, 1). We shall show thatP(G, t) has no zeros in (1, 32/27]. In addition, we shall construct graphs whose chromatic polynomials have zeros arbitrarily close to 32/27.

Hepatitis-Free Interval After Clotting Factor Therapy in First Infused Haemophiliacs

1986 ◽  
Vol 56 (03) ◽  
pp. 268-270 ◽  
Author(s):  
M Morfini ◽  
D Rafanelli ◽  
G Longo ◽  
A Messori ◽  
P Rossi Ferrini
Keyword(s):  
Life Table ◽  
Lower Risk ◽  
Clotting Factor ◽  
Rank Test ◽  
Life Table Method ◽  
Table Method ◽  
Log Rank Test ◽  
Free Interval ◽  
Post Infusion ◽  
Clotting Factor Concentrates

SummaryPost-infusion hepatitis is known to occur very frequently in haemophiliacs after treatment with unheated commercial clotting factor concentrates, obtained from large plasma donation pool. On the contrary, single-donor cryoprecipitate is likely to carry a lower risk of transmitting hepatitis.To evaluate this hypothesis, we retrospectively reviewed the medical records of 25 first infused haemophiliacs (from 1981 to 1984) treated with unheated commercial clotting factor concentrates (n = 19) or cryoprecipitate (n = 6).The hepatitis-free interval after the beginning of therapy was expressed as exposure days. The end point of each patient, i.e. the hepatitis occurrence, was defined as an increase of aminotransferases (ALT and AST) and/or the seroconversion of HBV-markers, which were checked every three months.The life-table method and log-rank test showed that cryo-precipitates had a significantly longer hepatitis-free interval (p = 0.0131, log-rank test) and a lower risk of transmitting hepatitis (p = 0.01-0.05, life-table method) than the commercial concentrates. However, the safety of cryoprecipitate therapy was shown to cover only a few exposure days, and so the real advantage of this product depends on the bleeding frequency of the patient concerned.We believe that these methods and our findings may be useful to assess and compare the safety of the new “heat-treated” clotting factor concentrates.

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line
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