Distinguishing tournaments with small label classes

A d-distinguishing vertex (arc) labeling of a digraph is a vertex (arc) labeling using d labels that is not preserved by any nontrivial automorphism. Let ρ(T) (ρ′(T)) be the minimum size of a label class in a 2-distinguishing vertex (arc) labeling of a tournament T. Gluck's Theorem implies that ρ(T) ≤ ⌊n/2⌋ for any tournament T of order n. We construct a family of tournaments ℌ such that ρ(T) ≥ ⌊n/2⌋ for any tournament of order n in ℌ. Additionally, we prove that ρ′(T) ≤ ⌊7n/36⌋ + 3 for any tournament T of order n and ρ′(T) ≥ ⌈n/6⌉ when T ∈ ℌ and has order n. These results answer some open questions stated by Boutin.Peer ReviewedPostprint (published version

Symmetry breaking in tournaments

We provide upper bounds for the determining number and the metric dimension of tournaments. A set of vertices S in V(T) is a determining set for a tournament T if every nontrivial automorphism of T moves at least one vertex of S, while S is a resolving set for T if every two distinct vertices in T have different distances to some vertex in S. We show that the minimum size of a determining set for an order n tournament (its determining number) is bounded by n/3, while the minimum size of a resolving set for an order n strong tournament (its metric dimension) is bounded by n/2. Both bounds are optimal.Peer ReviewedPostprint (published version

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Tournaments, 4-uniform hypergraphs, and an exact extremal result

We consider $4$-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of $5$ vertices spans either $0$ or exactly $2$ hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a `switching' operation on tournaments that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure