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+⋯+(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+⋯+(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 -