Statement Blair D . Sullivan 1 Past Work

In graduate school, my research has focused on problems in extremal directed graph theory. A 100-year old theorem of Mantel answers the question: what is the smallest value k so that, if every vertex in an undirected simple graph on n vertices has more than k edges coming out of it, the graph must contain a triangle? Mantel, and subsequently Turán, proved that this value is n/2. Their result is perhaps the earliest antecedent of extremal graph theory, a flourishing subdiscipline of combinatorics. Consider the corresponding question for a directed graph: what is the smallest value k so that, if every vertex in a simple∗ digraph on n vertices has at least k directed edges leaving it, the digraph must contain a directed triangle? In contrast with the case of undirected graphs, the answer to this problem is still unknown, despite having been proposed by Caccetta and Häggkvist to the mathematics community nearly thirty years ago. More specifically, given a directed graph G with vertex set V (G) and edge set E(G), say G is k-free if it has no directed cycle of length at most k. Also, let δ G denote the minimum outdegree in G. The formal statement is then: