Title: The Quest for the Mythical 57-regular Moore Graph: Removing Certain Classes of 4-cycles from a Partial Solution Abstract: Since 1968 an ongoing quest to find the hypothesized 57-regular Moore graph has been going on. Arguably the hardest existential question in graph theory, this problem has defeated both the increases in computing power and developments in graph theory since then. The problem statement is very simple: find an undirected regular graph of degree 57 having diameter 2 and girth 5. Diameter refers to the maximal distance between vertices, and girth refers to the smallest cycle length. If this graph exists, several of its properties are known and I will explain their derivation in the first part of this talk. For instance, the number of vertices must be 3250 and the number of edges 92625. The graph cannot be vertex-transitive and the order of its automorphism group is at most 375. As the layout of 3249 edges is already known, the second part of the talk will focus on the precise configuration of the remaining 89376 edges. I will explain how to remove all 3-cycles from a partial solution and how to classify all remaining 4-cycles in three categories A, B, C. I wrote a computer program that was able to remove both the 3-cycles and the 4-cycles of type A and B using a discrete optimization method. How to remove the C-class of 4-cycles is unknown at this point in time. Those who attend this talk can perhaps help by sharing their ideas about this problem or by relating the algorithm I will present to their own work in the field of discrete optimization.