Robust cycle bases do not exist for [formula omitted] if [formula omitted

Abstract

A basis for the cycle space of a graph is said to be robust if any cycle Z of G is a sum Z=C1+C2+⋯+Ck of basis elements such that (i) (C1+C2+⋯+Cℓ−1)∩Cℓ is a nontrivial path for each 2≤ℓ<k. Hence, (ii) each partial sum C1+C2+⋯+Cℓ is a cycle for 1≤ℓ≤k. While complete graphs and 2-connected plane graphs have robust cycle bases, it is shown that regular complete bipartite graphs Kn,n do not have any robust cycle basis if n≥8.