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A Continuous Analogue of the Girth Problem

Alon Amit, Shlomo Hoory, Nathan Linial
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  ... 
doi:10.1006/jctb.2001.2088 fatcat:if3no4idtvcepbzrthrgm4wfjy

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]

Reza Naserasr, Sagnik Sen, Qiang Sun
2015 arXiv   pre-print
As an extension 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.  ... 
arXiv:1501.05089v1 fatcat:4eiuhdueqrdjfih3q32l6gyirm

Notes on Cops and Robber game on graphs [article]

Bojan Mohar
2017 arXiv   pre-print
These are some personal notes about 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.  ... 
arXiv:1710.11281v1 fatcat:k63scaiaujchtknvxyqmhsgosm

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

Phillip E.C. Compeau
2011 Discrete Applied Mathematics  
We find that 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.  ... 
doi:10.1016/j.dam.2011.06.013 fatcat:igzjs46rwnfjljqd4qrhp3frei

Mixed Cages [article]

G. Araujo-Pardo, C. Hernández-Cruz, J.J. Montellano-Ballesteros
2017 arXiv   pre-print
We introduce 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.  ... 
arXiv:1702.07255v1 fatcat:dpk4c26zxrfdlbpk3cgll4ypbe

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

B P Reilly, M P Bolton, M J Lewis, L A Houghton, P J Whorwell
2002 Physiological Measurement  
We conclude that IP promises to be 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.  ... 
doi:10.1088/0967-3334/23/4/306 pmid:12450267 fatcat:ketfmtevmna5zjfm725rv5jkcq

The Distance-t Chromatic Index of Graphs

TOMÁŠ KAISER, ROSS J. KANG
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  ... 
doi:10.1017/s0963548313000473 fatcat:bvsulk4kmrc4zcwkoutjscgntu

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

Y. Rabinovich, R. Raz
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.  ... 
doi:10.1007/pl00009336 fatcat:htnftu6kpvhxtfg3kboitclkei

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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