Cycle construction and geodesic cycles with application to the hypercube

Abstract

Construction of cycles in a graph is investigated, where cycles from particular subsets (such as bases) are added together so that each partial sum is also a cycle or each new cycle intersects the sum of the preceding terms in a nontrivial path. Starting with the geodesic cycles, a hierarchical construction is given. For the hypercube graph, geodesic cycles are characterized, and it is shown how hypercube geodesic cycles can be constructed in two steps from a special basis. Applications are given to inferring commutativity of a diagram in a groupoid from commutativity of some of its cycles.

Canonical sphere bases for simplicial and cubical complexes

“…Proof. The result was proved for k = 1 in [28]; we now fix k ≥ 2 and proceed by induction on n; trivially, (i) and (ii) hold if n = k + 1. By the inductive hypothesis, for…”

“…The cycle basis B(n, 1) for the hypercube graph Q n has a weaker recursive property, called connected sum [28,19]: Any cycle in the graph can be constructed by iterating the procedure described for robust bases. The first iteration constructs a family of cycles from the original basis.…”

“…The simplex basis was known (though not its special property), while our cube basis seems to be new [28]. The simplex basis for k = 1 is robust [27] and the cube basis has the connected sum property [28]. See §5 below.…”

Classification of robust cycle bases and relations to fundamental cycle bases

“…Another line of research has been initiated by Kainen [6] who investigated robust cycle bases. Strengthening and weakening the concept of robust cycle bases led to four different types of robust cycle bases, which were further studied in [8] and [12], and recently in [7]. The latter paper provides an application of robust cycle bases to the analysis of commutative diagrams in groupoids.…”