Quantum annealing is a quantum algorithm for computing solutions to combinatorial optimization problems. This study proposes a method for minor embedding optimization problems onto sparse quantum annealing hardware graphs called 4-clique network minor embedding. This method is in contrast to the standard minor embedding technique of using a path of linearly connected qubits in order to represent a logical variable state. The 4-clique minor embedding is possible on Pegasus graph connectivity, which is the native hardware graph for some of the current D-Wave quantum annealers. The Pegasus hardware graph contains many cliques of size 4, making it possible to form a graph composed entirely of paths of connected 4-cliques on which a problem can be minor embedded. The 4-clique chains come at the cost of additional qubit usage on the hardware graph, but they allow for stronger coupling within each chain thereby increasing chain integrity, reducing chain breaks, and allow for greater usage of the available energy scale for programming logical problem coefficients on current quantum annealers. The 4-clique minor embedding technique is compared against the standard linear path minor embedding with experiments on two D-Wave quantum annealing processors with Pegasus hardware graphs. We show proof of concept experiments where the 4-clique minor embeddings can use weak chain strengths while successfully carrying out the computation of minimizing random all-to-all spin glass problem instances.