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Design theory is the subfield of combinatorics concerning the existence and construction of highly symmetric arrangements. Finite projective planes, latin squares, and Steiner triple systems are examples of designs.
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Is there a 7-regular graph on 50 vertices with girth 5? What about 57-regular on 3250 vertices?
The following problem is homework of a sort -- but homework I can't do!
The following problem is in Problem 1.F in Van Lint and Wilson:
Let $G$ be a graph where every vertex
has degree $d$. Su …