Non-isomorphic Extremal Graphs without Three-Cycles or Four-Cycles

Abstract

We define an $extremalgraph$ on $v$ vertices to be a graph that has the maximum number of edges on $v$ vertices, and that contains neither $3$-cycles nor $4$-cycles.
We establish that every vertex of degree at least $3$, in an extremal graph of at least $7$ vertices, is in a $5$-cycle; we enumerate all of the extremal graphs on $21$ or fewer vertices; and we determine the size of extremal graphs of orders $25$, $26$, and $27$.