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

2. 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)

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

4. How I met RJN

5. 18th BCC – University of Sussex, 2001

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

7. 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)

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

9. 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.

10. 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

11. Examples
Kn,n
Kn K2

12. But what about…

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

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–

15. An NCC graph

16. 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

17. 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

18. 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

19. 2009/10: The book…

20. Fran’s cake

21. Sketchy Tweets

22. Blog interview

23. “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

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