  There is a vast literature focused on network reliability evaluation. In the
last decades, reliability optimization has been also addressed. Frank Boesch in
1986 introduced the concept of uniformly most reliable graph (UMRG). Later,
Boesch \emph{et al.} presented the first UMRGs and conjectured that some
special subdivisions of the bipartite complete graph $K_{3,3}$, as well as the
bipartite complete graph $K_{4,4}$, are UMRGs. Wang proved that the first
conjecture is true. Wendy Myrvold confirmed that $K_{4,4}$ is also UMRG, by
means of computational tests. However, thus far, there is no mathematical proof
in the literature. A trivial cut is an edge-set that includes all the incident
edges of a fixed node. In this article we describe a methodology to determine
UMRGs based on bounding the number of trivial cuts. As a proof-of-concept it is
proved that both $K_{3,3}$ and $K_{4,4}$ are UMRGs.
