Long Paths in the Distance Graph over Large Subsets of Vector Spaces over Finite Fields

David Covert; Michael Bennett; Jonathan Chapman; Derrick Hart; Alex Iosevichc; Jonathan Pakianathan

doi:10.4134/JKMS.2016.53.1.115

Long Paths in the Distance Graph over Large Subsets of Vector Spaces over Finite Fields

David Covert
, Michael Bennett
, Jonathan Chapman
, Derrick Hart
, Alex Iosevichc
, Jonathan Pakianathan

University of Missouri-St. Louis

Research output: Contribution to journal › Article › peer-review

Abstract

<div class="line" id="line-5"> Let E⊂Fdq, the d-dimensional vector space over a finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x,y∈E by an edge if ||x−y||=(x1−y1)2+&ctdot;+(xd−yd)2=1. We shall prove that if the size of E is sufficiently large, then the distance graph of E contains long non-overlapping paths and vertices of high degree.</div>

Original language	American English
Journal	Journal of the Korean Mathematical Society
Volume	53
DOIs	https://doi.org/10.4134/JKMS.2016.53.1.115
State	Published - May 31 2014

Disciplines

Computer Sciences
Mathematics

Access to Document

10.4134/JKMS.2016.53.1.115License: Unspecified

LONG_PATHS_IN_THE_DISTANCE_GRAPH_OVER_LARGE_SUBSETS_OF_VECTOR_SPACES_OVER_FINITE_FIELDS (1).pdfFinal published version, 192 KBLicense: Unspecified

Cite this

@article{73f0f3c35b78492dbb46cf09aded5868,

title = "Long Paths in the Distance Graph over Large Subsets of Vector Spaces over Finite Fields",

abstract = " Let E\⊂Fdq, the d-dimensional vector space over a finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x,y\∈E by an edge if ||x\−y||=(x1\−y1)2+\&ctdot;+(xd\−yd)2=1. We shall prove that if the size of E is sufficiently large, then the distance graph of E contains long non-overlapping paths and vertices of high degree.",

author = "David Covert and Michael Bennett and Jonathan Chapman and Derrick Hart and Alex Iosevichc and Jonathan Pakianathan",

year = "2014",

month = may,

day = "31",

doi = "10.4134/JKMS.2016.53.1.115",

language = "American English",

volume = "53",

journal = "Journal of the Korean Mathematical Society",

}

TY - JOUR

T1 - Long Paths in the Distance Graph over Large Subsets of Vector Spaces over Finite Fields

AU - Covert, David

AU - Bennett, Michael

AU - Chapman, Jonathan

AU - Hart, Derrick

AU - Iosevichc, Alex

AU - Pakianathan, Jonathan

PY - 2014/5/31

Y1 - 2014/5/31

N2 - Let E⊂Fdq, the d-dimensional vector space over a finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x,y∈E by an edge if ||x−y||=(x1−y1)2+&ctdot;+(xd−yd)2=1. We shall prove that if the size of E is sufficiently large, then the distance graph of E contains long non-overlapping paths and vertices of high degree.

AB - Let E⊂Fdq, the d-dimensional vector space over a finite field with q elements. Construct a graph, called the distance graph of E, by letting the vertices be the elements of E and connect a pair of vertices corresponding to vectors x,y∈E by an edge if ||x−y||=(x1−y1)2+&ctdot;+(xd−yd)2=1. We shall prove that if the size of E is sufficiently large, then the distance graph of E contains long non-overlapping paths and vertices of high degree.

U2 - 10.4134/JKMS.2016.53.1.115

DO - 10.4134/JKMS.2016.53.1.115

M3 - Article

VL - 53

JO - Journal of the Korean Mathematical Society

JF - Journal of the Korean Mathematical Society

ER -