scholarly journals Average Degree Conditions Forcing a Minor

10.37236/5321 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Daniel J. Harvey ◽  
David R. Wood
Keyword(s):  
Recent Result ◽  
Complete Graph ◽  
Open Problem ◽  
Complete Bipartite Graph ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Arbitrary Graph ◽  
A Minor ◽  
High Degree

Mader first proved that high average degree forces a given graph as a minor. Often motivated by Hadwiger's Conjecture, much research has focused on the average degree required to force a complete graph as a minor. Subsequently, various authors have considered the average degree required to force an arbitrary graph $H$ as a minor. Here, we strengthen (under certain conditions) a recent result by Reed and Wood, giving better bounds on the average degree required to force an $H$-minor when $H$ is a sparse graph with many high degree vertices. This solves an open problem of Reed and Wood, and also generalises (to within a constant factor) known results when $H$ is an unbalanced complete bipartite graph.

scholarly journals The Degree-Diameter Problem for Sparse Graph Classes

10.37236/4313 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Guillermo Pineda-Villavicencio ◽  
David R. Wood
Keyword(s):  
Maximum Degree ◽  
Constant Factor ◽  
Average Degree ◽  
Sparse Graph ◽  
Sparse Graphs ◽  
Graph Classes ◽  
Minor Free Graphs

The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree $\Delta$ and diameter $k$. For fixed $k$, the answer is $\Theta(\Delta^k)$. We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is $\Theta(\Delta^{k-1})$, and for graphs of bounded arboricity the answer is $\Theta(\Delta^{\lfloor k/2\rfloor})$, in both cases for fixed $k$. For graphs of given treewidth, we determine the maximum number of vertices up to a constant factor. Other precise bounds are given for graphs embeddable on a given surface and apex-minor-free graphs.

scholarly journals Forcing a sparse minor

2015 ◽  
Vol 25 (2) ◽  
pp. 300-322 ◽  
Author(s):  
BRUCE REED ◽  
DAVID R. WOOD
Keyword(s):  
Complete Graph ◽  
Absolute Constant ◽  
Average Degree ◽  
Hadwiger's Conjecture ◽  
Minimum Number ◽  
A Minor ◽  
Hadwiger’S Conjecture

This paper addresses the following question for a given graphH: What is the minimum numberf(H) such that every graph with average degree at leastf(H) containsHas a minor? Due to connections with Hadwiger's conjecture, this question has been studied in depth whenHis a complete graph. Kostochka and Thomason independently proved that$f(K_t)=ct\sqrt{\ln t}$. More generally, Myers and Thomason determinedf(H) whenHhas a super-linear number of edges. We focus on the case whenHhas a linear number of edges. Our main result, which complements the result of Myers and Thomason, states that ifHhastvertices and average degreedat least some absolute constant, then$f(H)\leq 3.895\sqrt{\ln d}\,t$. Furthermore, motivated by the case whenHhas small average degree, we prove that ifHhastvertices andqedges, thenf(H) ⩽t+ 6.291q(where the coefficient of 1 in thetterm is best possible).

scholarly journals A Lower Bound on the Average Degree Forcing a Minor

10.37236/8847 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Sergey Norin ◽  
Bruce Reed ◽  
Andrew Thomason ◽  
David R. Wood
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Constant Factor ◽  
Average Degree ◽  
Complete Graphs ◽  
Sparse Graphs ◽  
Dense Graphs ◽  
A Minor

We show that for sufficiently large $d$ and for $t\geq d+1$,  there is a graph $G$ with average degree $(1-\varepsilon)\lambda  t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete graphs by Thomason (2001) and for general dense graphs by Myers and Thomason (2005). It also shows that an upper bound for sparse graphs by Reed and Wood (2016) is best possible up to a constant factor.

scholarly journals A Note on Pair Sum Graphs

2011 ◽  
Vol 3 (2) ◽  
pp. 321-329 ◽  
Author(s):  
R. Ponraj ◽  
J. X. V. Parthipan ◽  
R. Kala
Keyword(s):  
Bipartite Graph ◽  
Complete Graph ◽  
Complete Bipartite Graph ◽  
Edge Function ◽  
Injective Map ◽  
Cycle Path

Let G be a (p,q) graph. An injective map ƒ: V (G) →{±1, ±2,...,±p} is called a pair sum labeling if the induced edge function, ƒe: E(G)→Z -{0} defined by ƒe (uv)=ƒ(u)+ƒ(v) is one-one and ƒe(E(G)) is either of the form {±k1, ±k2,…, ±kq/2} or {±k1, ±k2,…, ±k(q-1)/2} {k (q+1)/2} according as q is even or odd. Here we prove that every graph is a subgraph of a connected pair sum graph. Also we investigate the pair sum labeling of some graphs which are obtained from cycles. Finally we enumerate all pair sum graphs of order ≤ 5.Keywords: Cycle; Path; Bistar; Complete graph; Complete bipartite graph; Triangular snake.© 2011 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi:10.3329/jsr.v3i2.6290                 J. Sci. Res. 3 (2), 321-329 (2011)

scholarly journals INTRINSICALLY LINKED GRAPHS WITH KNOTTED COMPONENTS

2012 ◽  
Vol 21 (07) ◽  
pp. 1250065 ◽  
Author(s):  
THOMAS FLEMING
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Linking Number ◽  
A Minor

We construct a graph G such that any embedding of G into R3 contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H contains a nonsplit n component link, where at least m of the components are nontrivial knots. We then turn our attention to complete graphs and show that for any given n, every embedding of a large enough complete graph contains a 2-component link whose linking number is a nonzero multiple of n. Finally, we show that if a graph is a Cartesian product of the form G × K2, it is intrinsically linked if and only if G contains one of K5, K3,3 or K4,2 as a minor.

k-Degenerate Graphs

1970 ◽  
Vol 22 (5) ◽  
pp. 1082-1096 ◽  
Author(s):  
Don R. Lick ◽  
Arthur T. White
Keyword(s):  
Bipartite Graph ◽  
Chromatic Number ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Natural Numbers ◽  
Image Position ◽  
Disconnected Graphs ◽  
Totally Disconnected

Graphs possessing a certain property are often characterized in terms of a type of configuration or subgraph which they cannot possess. For example, a graph is totally disconnected (or, has chromatic number one) if and only if it contains no lines; a graph is a forest (or, has point-arboricity one) if and only if it contains no cycles. Chartrand, Geller, and Hedetniemi [2] defined a graph to have property Pn if it contains no subgraph homeomorphic from the complete graph Kn+1 or the complete bipartite graphFor the first four natural numbers n, the graphs with property Pn are exactly the totally disconnected graphs, forests, outerplanar and planar graphs, respectively. This unification suggested the extension of many results known to hold for one of the above four classes of graphs to one or more of the remaining classes.

scholarly journals River landscapes and optimal channel networks

2018 ◽  
Vol 115 (26) ◽  
pp. 6548-6553 ◽  
Author(s):  
Paul Balister ◽  
József Balogh ◽  
Enrico Bertuzzo ◽  
Béla Bollobás ◽  
Guido Caldarelli ◽  
...  
Keyword(s):  
Minimum Energy ◽  
Height Function ◽  
Gravitational Energy ◽  
Constant Factor ◽  
Channel Networks ◽  
Arbitrary Graph ◽  
Tree Structures ◽  
Dimensional Lattice ◽  
Natural Forms ◽  
Flow Directions

We study tree structures termed optimal channel networks (OCNs) that minimize the total gravitational energy loss in the system, an exact property of steady-state landscape configurations that prove dynamically accessible and strikingly similar to natural forms. Here, we show that every OCN is a so-called natural river tree, in the sense that there exists a height function such that the flow directions are always directed along steepest descent. We also study the natural river trees in an arbitrary graph in terms of forbidden substructures, which we call k-path obstacles, and OCNs on a d-dimensional lattice, improving earlier results by determining the minimum energy up to a constant factor for everyd≥2. Results extend our capabilities in environmental statistical mechanics.

Classification of distance-transitive orbital graphs of overgroups of the Jevons group

2020 ◽  
Vol 30 (1) ◽  
pp. 7-22
Author(s):  
Boris A. Pogorelov ◽  
Marina A. Pudovkina
Keyword(s):  
Vector Space ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Isometry Group ◽  
Complete Bipartite Graph ◽  
Dimensional Vector ◽  
Translation Group ◽  
Dimensional Vector Space ◽  
Hamming Metric

AbstractThe Jevons group AS̃n is an isometry group of the Hamming metric on the n-dimensional vector space Vn over GF(2). It is generated by the group of all permutation (n × n)-matrices over GF(2) and the translation group on Vn. Earlier the authors of the present paper classified the submetrics of the Hamming metric on Vn for n ⩾ 4, and all overgroups of AS̃n which are isometry groups of these overmetrics. In turn, each overgroup of AS̃n is known to define orbital graphs whose “natural” metrics are submetrics of the Hamming metric. The authors also described all distance-transitive orbital graphs of overgroups of the Jevons group AS̃n. In the present paper we classify the distance-transitive orbital graphs of overgroups of the Jevons group. In particular, we show that some distance-transitive orbital graphs are isomorphic to the following classes: the complete graph 2n, the complete bipartite graph K2n−1,2n−1, the halved (n + 1)-cube, the folded (n + 1)-cube, the graphs of alternating forms, the Taylor graph, the Hadamard graph, and incidence graphs of square designs.

Recycling Scrap Crosslinked Urethane Polymers

1975 ◽  
Vol 48 (5) ◽  
pp. 972-980 ◽  
Author(s):  
R. M. Gerkin ◽  
F. E. Critchfield ◽  
W. A. Miller ◽  
R. Roberts ◽  
C. G. Seefried
Keyword(s):  
Particle Size ◽  
Ambient Temperature ◽  
Size Reduction ◽  
Minor Effect ◽  
Particle Size Reduction ◽  
Foam Formulation ◽  
A Minor ◽  
High Degree ◽  
Sulfur Curing

Abstract A. Scrap LRM polymers can be ground to powder on the Banbury operating at ambient temperature with cooling of the rotors and jacket. Particle size reduction can be accomplished in 5 min at 180 rpm. B. Powdered LRM polymer can be blended with TPU up to 50 wt.% and the composite compression molded to give the same properties as the TPU. C. Powdered LRM polymer can be blended with nitrile, chloroprene, and EPDM rubbers to give incompatible composites. The powdered LRM polymer acts in a manner similar to typical nonreinforcing fillers. D. It is possible to degrade powdered LRM polymers to a tacky mass at 180°C, which behaves much like a typical extender oil, when blended with nitrile and chloroprene rubbers. It is speculated that the degraded urethane should show a high degree of permanency in such blends. E. Addition of the urethane polymer to the rubber stocks does not interfere with the standard sulfur-curing mechanisms. F. Scrap HR foam can be ground to a powder in the Banbury in a manner similar to LRM scrap. G. A blend of 5 phr of powdered foam with a new HR foam formulation was machine processable. Addition of the powder had only a minor effect on the properties of the new foam.

scholarly journals Predator-induced flow disturbances alert prey, from the onset of an attack

2014 ◽  
Vol 281 (1790) ◽  
pp. 20141083 ◽  
Author(s):  
Jérôme Casas ◽  
Thomas Steinmann
Keyword(s):  
Prey Capture ◽  
Ground Effect ◽  
Just In Time ◽  
Soil Arthropods ◽  
Predator Prey ◽  
Flow Sensing ◽  
Predator Attack ◽  
Flow Disturbances ◽  
A Minor ◽  
High Degree

Many prey species, from soil arthropods to fish, perceive the approach of predators, allowing them to escape just in time. Thus, prey capture is as important to predators as prey finding. We extend an existing framework for understanding the conjoint trajectories of predator and prey after encounters, by estimating the ratio of predator attack and prey danger perception distances, and apply it to wolf spiders attacking wood crickets. Disturbances to air flow upstream from running spiders, which are sensed by crickets, were assessed by computational fluid dynamics with the finite-elements method for a much simplified spider model: body size, speed and ground effect were all required to obtain a faithful representation of the aerodynamic signature of the spider, with the legs making only a minor contribution. The relationship between attack speed and the maximal distance at which the cricket can perceive the danger is parabolic; it splits the space defined by these two variables into regions differing in their values for this ratio. For this biological interaction, the ratio is no greater than one, implying immediate perception of the danger, from the onset of attack. Particular attention should be paid to the ecomechanical aspects of interactions with such small ratio, because of the high degree of bidirectional coupling of the behaviour of the two protagonists. This conclusion applies to several other predator–prey systems with sensory ecologies based on flow sensing, in air and water.

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