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Title Proposer Category
Extremal Graphs with Small Girth
 Michael Codish Alice Miller Patrick Prosser Peter Stuckey
Combinatorial mathematics

Given a simple undirected graph \$G = (V,E)\$, where \$V\$ is the set of vertices and \$E\$ the set of undirected edges, the edge {\$u,v\$} is in \$E\$ if and only if vertex \$u\$ is adjacent to vertex \$v\$ in \$G\$. The graph is simple in that there are no loop edges, i.e. we have no edges of the form {\$v,v\$}.

The order (\$m\$) of a graph is the number of vertices in that graph and the size (\$n\$) is the number of edges. The girth (\$k\$) is the length of the smallest cycle contained in the graph.

Let \$fk(m)\$ denote the maximum number of edges in a graph with m vertices and girth greater than \$k\$ (i.e. with no cycles of length \$k\$ or less). The number of non-isomorphic extremal graphs with m vertices and \$fk(m)\$ edges is denoted \$Fk(m)\$.

The problem is then, for given values of \$m\$ and \$k\$, find \$fk(m)\$ and additionally to find \$Fk(m)\$. That is, what is the maximal number of edges in a graph with m vertices and girth greater than \$k\$ and how many such unique graphs are there?