Long paths and cycles in subgraphs of the cube

Details Long paths and cycles in subgraphs of the cube Long paths and cycles in subgraphs of the cube

2010-06-15

arxiv.org/abs/1006.3049v1

Mathematics

Combinatorics

24 pages, 6 figures

Scientific paper

Let $Q_n$ denote the graph of the $n$-dimensional cube with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. Suppose $G$ is a subgraph of $Q_n$ with average degree at least $d$. How long a path can we guarantee to find in $G$? Our aim in this paper is to show that $G$ must contain an exponentially long path. In fact, we show that if $G$ has minimum degree at least $d$ then $G$ must contain a path of length $2^d-1$. Note that this bound is tight, as shown by a $d$-dimensional subcube of $Q_n$. We also obtain the slightly stronger result that $G$ must contain a cycle of length at least $2^d$.

Affiliated with

Also associated with

No associations

Rating

Long paths and cycles in subgraphs of the cube does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Long paths and cycles in subgraphs of the cube, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Long paths and cycles in subgraphs of the cube will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-105472