Games and graphs: the legacy of RJN CanaDAM’17 Games and graphs: the legacy of RJN Anthony Bonato Ryerson University

RJN-ology 130+ publications in GT and CGT ~3K citations books: Lessons in Play The Game of Cops and Robbers on Graphs several edited volumes 12+ doctoral students, 16+ Masters, many more undergraduate RAs/theses University Research Professor (Dal) Adrien Pouliot Award (CMS)

What I learned from RJN Pose good problems. Work with students. Do more than one thing. Play.

How I met RJN

18th BCC – University of Sussex, 2001

R V = primes congruent to 1 (mod 4) E: pq an edge if 𝑝 𝑞 =1 undirected by quadratic reciprocity

Infinite random graph G(N,1/2): V = N E: sample independently with probability ½ Theorem (Erdős,Rényi,63) With probability 1, two graphs sampled from G(N,1/2) are isomorphic to R. holds also for any fixed p ∈(0,1)

Properties of R diameter 2 universal indestructible indivisible pigeonhole property axiomatizes almost sure theory of graphs …

Local properties of R Neighborhood Property (N): R has (N) For each vertex x, each of the subgraphs induced by N(x) and Nc(x) are isomorphic to G R has (N) Question posed by me at BCC’18 problem session: Do any other graphs have (N)? Yes! P. Gordinowicz, On graphs isomorphic to their neighbour and non-neighbour sets, European J. Combin. 31 (2010) 1419–1428.

RJN and NCC no finite graph has (N) only finite graph with subgraphs induced by N(x) and Nc(x) isomorphic for all x is K1 (exercise) RJN suggested instead: (NCC): For all vertices x, the subgraphs induced by N(x) and Nc[x] are isomorphic

Examples Kn,n Kn K2

But what about…

B, RJN, Partitioning a graph into two isomorphic pieces, Journal of Graph Theory 44 (2003) 1-14.

Characterization dnp matching: perfect matching M such that for all e = ab ∈ M, the neighbour sets of a and b are disjoint Theorem (B,RJN,03): A graph G is NCC iff there is a positive integer n such that: G is order 2n; G is n-regular; and G has a dnp matching. checking if G is NCC is in P: non-existence of Tutte sets in G–

An NCC graph

Other directions (Priesler-Moreno,05) another characterization of NCC graphs via GNCC graphs. (B,06) characterized spanning subgraphs of NCC graphs also in P (B,Costea,06) locally H-perfect matchings, H = C4, P4, paw, diamond

Open problems E(x) and O(x): subgraphs induced by vertices of even and odd distance, resp. Problem: characterize graphs G with NCC-e: such that for all x, E(X) and O(x) are isomorphic. examples: NCC graphs (which are diameter 2) balanced bipartite graphs

Open problems Problem: Characterize infinite NCC graphs α-regular for infinite cardinal α exist α-regular graphs with a dnp matching that are not NCC

2009/10: The book…

Fran’s cake

Sketchy Tweets

Blog interview

“I think students are very important and the best ones are self-motivated. They have to have confidence, not necessarily a great background. They need confidence to stand up to me and say I’m wrong. It’s one of the reasons I like games; if you play a game by yourself it gets really boring.” -RJN

Thank you, RJN. Keep playing. All the best on your infinite sabbatical.