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)
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What I learned from RJN
• Pose good problems.
• Work with students.
• Do more than one thing.
• Play.
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How I met RJN
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18th BCC – University of Sussex, 2001
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R
• V = primes congruent to 1 (mod 4)
• E: pq an edge if
𝑝
𝑞
=1
– undirected by quadratic reciprocity
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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)
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Properties of R
• diameter 2
• universal
• indestructible
• indivisible
• pigeonhole property
• axiomatizes almost sure theory of graphs
…
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Local properties of R
• Neighborhood Property (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.
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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
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Examples
• Kn,n
• Kn K2
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But what about…
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B, RJN, Partitioning a graph into two
isomorphic pieces,
Journal of Graph Theory 44 (2003) 1-14.
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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:
1. G is order 2n;
2. G is n-regular; and
3. G has a dnp matching.
• checking if G is NCC is in P:
– non-existence of Tutte sets in G–
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An NCC graph
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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
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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
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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
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2009/10: The book…
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Fran’s cake
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Sketchy Tweets
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Blog interview
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“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
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Thank you, RJN.
Keep playing.
All the best on your infinite sabbatical.
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