Counting graph orientations with no directed triangles
Pedro Ara\'ujo, F\'abio Botler, Guilherme Oliveira Mota

TL;DR
This paper confirms a conjecture by Alon and Yuster regarding the maximum number of orientations of an n-vertex graph with no directed triangles, establishing the extremal structure as the balanced complete bipartite graph.
Contribution
It proves the conjecture that the maximum is achieved for sufficiently large n and characterizes the unique extremal graphs achieving this maximum.
Findings
Maximum number of orientations is $2^{loor{n^2/4}}$ for large n.
Balanced complete bipartite graph uniquely extremal.
Confirms the conjecture for all n ≥ 8.
Abstract
Alon and Yuster proved that the number of orientations of any -vertex graph in which every is transitively oriented is at most for and conjectured that the precise lower bound on should be . We confirm their conjecture and, additionally, characterize the extremal families by showing that the balanced complete bipartite graph with vertices is the only -vertex graph for which there are exactly such orientations.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph Labeling and Dimension Problems · Advanced Graph Theory Research
