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A Continuous Analogue of the Girth Problem

2002
*
Journal of combinatorial theory. Series B (Print)
*

*The*best known bound for this

*problem*is due to Biggs [2] . In Section 5 we establish

*a*bound on

*the*distributional

*analogue*

*of*

*the*same

*problem*. ... Since D=W(log n),

*the*best known lower bound on L as

*a*function

*of*n and d is therefore: L \ 2`d − 1 1 1 − O 1 1 log n 2 22 . (17)

*GIRTH*

*PROBLEM*Here, we consider

*a*distributional

*analogue*(called L D ...

##
###
Page 598 of Mathematical Reviews Vol. , Issue 93b
[page]

1993
*
Mathematical Reviews
*

*The*advantage

*of*this transformation

*of*

*the*

*problem*is that

*the*rank-sum maximization

*problem*for ideals

*of*fixed size in Limon has

*a*

*continuous*

*analogue*solvable by variational means. ... ,

*the*

*continuous*

*analogue*

*of*

*the*transformed Kleitman-West

*problem*is to maximize r(S) = f, r(x) dx over all lower ideals S C L, having volume /’, where r(x) = >> x;. ...

##
###
Page 8809 of Mathematical Reviews Vol. , Issue 2002M
[page]

2002
*
Mathematical Reviews
*

) ;
Linial, Nathan (IL-HEBR-IC; Jerusalem)

*A**continuous**analogue**of**the**girth**problem*. ...*The**girth**of**a*graph is*the*length*of**the*shortest cycle it contains. If we require that*a*graph has*girth*g and is regular*of*degree r, what is*the*fewest number*of*vertices that it can contain? ...##
###
Walk-powers and homomorphism bound of planar graphs
[article]

2015
*
arXiv
*
pre-print

As an extension

arXiv:1501.05089v1
fatcat:4eiuhdueqrdjfih3q32l6gyirm
*of**the*Four-Color Theorem it is conjectured that every planar graph*of*odd-*girth*at least 2k+1 admits*a*homomorphism to PC_2k=(Z_2^2k, {e_1, e_2, ... ... Noting that PC_2k itself is*of*odd-*girth*2k+1, in this work we show that if*the*conjecture is true, then PC_2k is an optimal such*a*graph both with respect to number*of*vertices and number*of*edges. ... This conjecture, for odd values*of*d was introduced in [N07] in relation to*a*question*of*J. ...##
###
Notes on Cops and Robber game on graphs
[article]

2017
*
arXiv
*
pre-print

These are some personal notes about

arXiv:1710.11281v1
fatcat:k63scaiaujchtknvxyqmhsgosm
*the*pursuit game*of*Cops and Robbers that I made starting in 2007. More old and new*problems*(and some solutions) will be added in future versions*of*these notes. ... This strategy works well in*the**continuous*and in*the*discrete version. Lemma 3 . 3 If*girth*(G) ≥ 5, then c(G) ≥ δ(G). ...*The*natural question arises what is*the*largest d = d(g) for which there is*a*graph*of*genus at most g and*girth*at least 5. ...##
###
Page 272 of Mathematical Reviews Vol. 50, Issue 2
[page]

1975
*
Mathematical Reviews
*

This is

*a**continuation**of**the*article reviewed above [#1946].*A*corollary*of**the*theorem by Harary and Norman is proved in*the*first part*of**the*article. ...*The*proof*of*an*analogue**of*Otter’s formula for g(u,*a*) is*the*second result. L. S. Mel'nikov (Novosibirsk) Cook, R. J. Heawood’s theorem and connectivity. Mathematika 20 (1973), 201-207. ...##
###
Girth of pancake graphs

2011
*
Discrete Applied Mathematics
*

We find that

doi:10.1016/j.dam.2011.06.013
fatcat:igzjs46rwnfjljqd4qrhp3frei
*the**girth**of*each family*of*pancake graphs remains constant after some small threshold value*of*n. ... We consider four families*of*pancake graphs, which are Cayley graphs, whose vertex sets are either*the*symmetric group on n objects or*the*hyperoctahedral group on n objects and whose generating sets are ... Acknowledgments*The*author would like to thank Dr. Laurie Heyer for her guidance,*the*referees for insightful comments, and*the*Davidson Research Initiative for its support. ...##
###
Mixed Cages
[article]

2017
*
arXiv
*
pre-print

We introduce

arXiv:1702.07255v1
fatcat:dpk4c26zxrfdlbpk3cgll4ypbe
*the*notion*of**a*[z, r; g]-mixed cage.*A*[z, r; g]-mixed cage is*a*mixed graph G, z-regular by arcs, r-regular by edges, with*girth*g and minimum order. ... In this paper we prove*the*existence*of*[z, r ;g]-mixed cages and exhibit families*of*mixed cages for some specific values. We also give lower and upper bounds for some choices*of*z, r and g. ... Acknowledgment*The*authors wish to thank*the*anonymous referees*of*this paper. ...##
###
Page 1769 of Mathematical Reviews Vol. 51, Issue 6
[page]

1976
*
Mathematical Reviews
*

*A*representative arrangement 4 is

*continuous*if

*the*directional coordinates

*of*

*the*lines

*of*

*A*are

*continuous*functions on

*the*sphere S¢-?. ... Another challenging suggestion is to find

*a*tactical configuration

*analogue*

*of*

*the*Bruck-Ryser- Chowla theorem on block designs. ...

##
###
A device for 24 hour ambulatory monitoring of abdominal girth using inductive plethysmography

2002
*
Physiological Measurement
*

We conclude that IP promises to be

doi:10.1088/0967-3334/23/4/306
pmid:12450267
fatcat:ketfmtevmna5zjfm725rv5jkcq
*a*useful and quantitative tool suitable for ambulatory monitoring*of*abdominal*girth*,*a*hitherto relatively unexplored symptom*of*IBS. ...*The*system was separately calibrated by placing*the*belts around*a*variable rectangular phantom and measuring*the*frequency*of*oscillation. ... IP therefore promised to be*a*solution, possibly without some*of**the**problems*encountered with*the*respiration work. ...##
###
The Distance-t Chromatic Index of Graphs

2013
*
Combinatorics, probability & computing
*

*The*other is

*a*bound

*of*O(Δ^t/Δ) (as Δ→∞) for graphs

*of*maximum degree at most Δ and

*girth*at least 2t+1.

*The*first bound is an

*analogue*

*of*Molloy and Reed's bound on

*the*strong chromatic index. ...

*The*second bound is tight up to

*a*constant multiplicative factor, as certified by

*a*class

*of*graphs

*of*

*girth*at least g, for every fixed g > 3,

*of*arbitrarily large maximum degree Δ, with distance-t chromatic ... Thus, it remains to prove

*the*inequality (5). i:x i ≥

*a*/2 x i > an 2 . (6) v e * Bt W

*A*t−1

*A*1 ≥ ∆/2 . . . u Before

*continuing*, let us make

*a*basic observation about

*the*cardinality

*of*

*A*: (2 − 2δ)∆ t−1 ...

##
###
Page 51 of Problems in General Surgery Vol. 15, Issue 1
[page]

1998
*
Problems in General Surgery
*

Symptoms

*of**a*distinct episode*of*acute pancreatitis are seldom elicited, and patients usually have symptoms*of*increasing abdominal*girth*, weight loss, and muscle wast- ing. ... PANCREATIC ASCITES Ascites may occur when pancreatic secre- tions collect within*the*abdomen after disrup- tion*of**the*pancreatic duct or when*a*pseudocyst*continuously*leaks into*the*abdomen. ...##
###
Lower Bounds on the Distortion of Embedding Finite Metric Spaces in Graphs

1998
*
Discrete & Computational Geometry
*

*The*main technical lemma claims that

*the*metric space induced by an unweighted graph H

*of*

*girth*g cannot be embedded in

*a*graph G (even if it is weighted)

*of*smaller Euler characteristic, with distortion ... In addition, we discuss

*the*case χ(G) < χ(H) − 1, as well as some interesting higher-dimensional

*analogues*.

*The*proofs employ basic techniques

*of*algebraic topology. 80 Y. Rabinovich and R. ... Yuri Rabinovich would also like to thank Nathan Linial for discussion

*of*

*the*initial questions that led to this paper. ...

##
###
Page 3586 of Mathematical Reviews Vol. , Issue 83i
[page]

1983
*
Mathematical Reviews
*

*The*main part

*of*

*the*paper is

*the*proof

*of*

*a*combinatorial theorem that gives

*a*lower bound for

*the*maximal size

*of*an independent set in

*a*3-graph with

*girth*greater than 4. ...

*A*one-sided

*analogue*

*of*

*the*Lagrange spectrum is obtained as

*the*range

*of*

*the*function

*a*(@)=limsup{1/q( p—q@):p €Z,qEN, p—q@>0}, defined for each real number @. ...

##
###
Page 4746 of Mathematical Reviews Vol. , Issue 82k
[page]

1982
*
Mathematical Reviews
*

*continuity*. ... From

*the*introduction: “In this note we examine some questions involving

*the*

*girth*

*of*

*the*unit sphere

*of*

*a*normed space—

*the*infimum

*of*

*the*lengths

*of*symmetric simple closed curves on

*the*sphere. ...

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