On k-Connected Subgraphs of the Hypercube

Abstract

This chapter focuses on k-connected subgraphs of the hypercube. The hypercube of dimension n , denoted by Q n , is the graph of 2 n vertices labeled by binary vectors of length n , an edge joining two vertices whenever the corresponding vectors differ in exactly one coordinate. A C -valuation of the graph G is its edge coloring such that (1) in each cycle no color occurs an odd number of times and (2) in each open path there is a color that occurs an odd number of times. A graph is cubical if there exists a C -valuation. By a C n -valuation of G , it is understood that in the C -valuation of G that exactly n colors are used. The smallest n for which G is a subgraph of Q n is called the “cubical dimension” of G and denoted by cd( G ). If G is connected, cd( G ) is the smallest n such that there exists a C -valuation of G (the color of an edge in the C -valuation corresponds to the coordinate in which its endvertices differ).