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.

scholarly journals Chromatic zeros and the golden ratio

2009 ◽  
Vol 3 (1) ◽  
pp. 120-122 ◽  
Author(s):  
Saeid Alikhani ◽  
Yee-Hock Peng
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Chromatic Polynomial ◽  
Chromatic Roots

In this note, we investigate ?n , where ?=1+2?5 is the golden ratio as chromatic roots. Using some properties of Fibonacci numbers, we prove that ? n (n ? N), cannot be roots of any chromatic polynomial.

scholarly journals Chromatic Bounds on Orbital Chromatic Roots

10.37236/4412 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dae Hyun Kim ◽  
Alexander H. Mun ◽  
Mohamed Omar
Keyword(s):  
Positive Integer ◽  
Real Root ◽  
Chromatic Polynomial ◽  
Outerplanar Graphs ◽  
Chromatic Polynomials ◽  
The Real ◽  
Real Roots ◽  
Chromatic Roots

Given a group $G$ of automorphisms of a graph $\Gamma$, the orbital chromatic polynomial $OP_{\Gamma,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $\Gamma.$ Cameron and Kayibi introduced this polynomial as a means of understanding roots of chromatic polynomials. In this light, they posed a problem asking whether the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root of its chromatic polynomial. We resolve this problem in a resounding negative by not only constructing a counterexample, but by providing a process for generating families of counterexamples. We additionally begin the program of finding classes of graphs whose orbital chromatic polynomials have real roots bounded above by the largest real root of their chromatic polynomials; in particular establishing this for many outerplanar graphs.

scholarly journals KNOT-INEVITABLE PROJECTIONS OF PLANAR GRAPHS

1996 ◽  
Vol 05 (06) ◽  
pp. 877-883 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
TATSUYA TSUKAMOTO
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Spatial Graph ◽  
Torus Knot ◽  
Nonplanar Graph

For each odd number n, we describe a regular projection of a planar graph such that every spatial graph obtained by giving it over/under information of crossing points contains a (2, n)-torus knot. We also show that for any spatial graph H, there is a regular projection of a (possibly nonplanar) graph such that every spatial graph obtained from it contains a subgraph that is ambient isotopic to H.

r-Hued coloring of planar graphs without short cycles

2020 ◽  
Vol 12 (03) ◽  
pp. 2050034
Author(s):  
Yuehua Bu ◽  
Xiaofang Wang
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Minimum Integer

A [Formula: see text]-hued coloring of a graph [Formula: see text] is a proper [Formula: see text]-coloring [Formula: see text] such that [Formula: see text] for any vertex [Formula: see text]. The [Formula: see text]-hued chromatic number of [Formula: see text], written [Formula: see text], is the minimum integer [Formula: see text] such that [Formula: see text] has a [Formula: see text]-hued coloring. In this paper, we show that [Formula: see text] if [Formula: see text] and [Formula: see text] is a planar graph without [Formula: see text]-cycles or if [Formula: see text] is a planar graph without [Formula: see text]-cycles and no [Formula: see text]-cycle is intersect with [Formula: see text]-cycles, [Formula: see text], then [Formula: see text], where [Formula: see text].

scholarly journals Planar Graphs of Maximum Degree Six without 7-cycles are Class One

10.37236/2589 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Danjun Huang ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

In this paper, we prove that every planar graph of maximum degree six without 7-cycles is class one.

scholarly journals Densities of Minor-Closed Graph Families

10.37236/408 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
David Eppstein
Keyword(s):  
Order Type ◽  
Closed Graph ◽  
Simple Graphs ◽  
Minimal Graphs ◽  
Vertex Graph ◽  
Graph Families ◽  
A Minor ◽  
Cluster Points

We define the limiting density of a minor-closed family of simple graphs $\mathcal{F}$ to be the smallest number $k$ such that every $n$-vertex graph in $\mathcal{F}$ has at most $kn(1+o(1))$ edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least $\omega^\omega$. It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, $1$ and $3/2$. For multigraphs, the only possible limiting densities are the integers and the superparticular ratios $i/(i+1)$.

scholarly journals Planar Graphs have Independence Ratio at least 3/13

10.37236/5309 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Disjoint Union ◽  
Independence Number ◽  
Independent Set

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$.  In 1968, Erdős asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$.

scholarly journals Algebraic Properties of Chromatic Roots

10.37236/6578 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Peter J. Cameron ◽  
Kerri Morgan
Keyword(s):  
Algebraic Integer ◽  
Tutte Polynomial ◽  
Chromatic Polynomial ◽  
Galois Groups ◽  
Algebraic Numbers ◽  
Isaac Newton ◽  
Isaac Newton Institute ◽  
Chromatic Root ◽  
The Subject ◽  
Chromatic Roots

A chromatic root is a root of the chromatic polynomial of a graph.  Any chromatic root is an algebraic integer. Much is known about the location of chromatic roots in the real and complex numbers, but rather less about their properties as algebraic numbers. This question was the subject of a seminar at the Isaac Newton Institute in late 2008.  The purpose of this paper is to report on the seminar and subsequent developments.We conjecture that, for every algebraic integer $\alpha$, there is a natural number n such that $\alpha+n$ is a chromatic root. This is proved for quadratic integers; an extension to cubic integers has been found by Adam Bohn. The idea is to consider certain special classes of graphs for which the chromatic polynomial is a product of linear factors and one "interesting" factor of larger degree. We also report computational results on the Galois groups of irreducible factors of the chromatic polynomial for some special graphs. Finally, extensions to the Tutte polynomial are mentioned briefly.

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Edge Coloring ◽  
Discrete Math ◽  
Adjacent Vertex ◽  
Chromatic Index ◽  
Proper Edge Coloring ◽  
Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

A Important Property on Planar Graph

2011 ◽  
Vol 50-51 ◽  
pp. 245-248
Author(s):  
Na Na Li ◽  
Shu Jun Wang ◽  
Xian Rui Meng
Keyword(s):  
Planar Graph ◽  
Planar Graphs

In 1994, C.Thomassen proved that every planar graph is (namely ). In 1993, M.Voigt shown that there are planar graphs which are not . But no one know whether every planar graph is . In this paper, we give a important property on planar graph that “Every planar graph is ” and “Every planar graph is free ” are equivalent.

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