graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, Conway's 99-graph problem is an unsolved problem asking whether there exists an

undirected graph In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also call ...

with 99 vertices, in which each two adjacent vertices have exactly one common neighbor, and in which each two non-adjacent vertices have exactly two common neighbors. Equivalently, every edge should be part of a unique triangle and every non-adjacent pair should be one of the two diagonals of a unique 4-cycle.

John Horton Conway John Horton Conway (26 December 1937 – 11 April 2020) was an English mathematician. He was active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to many b ...

offered a $1000 prize for its solution.

Properties

If such a graph exists, it would necessarily be a

locally linear graph In graph theory, a locally linear graph is an undirected graph in which every edge belongs to exactly one triangle. Equivalently, for each vertex of the graph, its neighborhood (graph theory), neighbors are each adjacent to exactly one other neigh ...

and a

strongly regular graph In graph theory, a strongly regular graph (SRG) is a regular graph with vertices and degree such that for some given integers \lambda, \mu \ge 0 * every two adjacent vertices have common neighbours, and * every two non-adjacent vertices h ...

with parameters (99,14,1,2). The first, third, and fourth parameters encode the statement of the problem: the graph should have 99 vertices, every pair of adjacent vertices should have 1 common neighbor, and every pair of non-adjacent vertices should have 2 common neighbors. The second parameter means that the graph is a

regular graph In graph theory, a regular graph is a Graph (discrete mathematics), graph where each Vertex (graph theory), vertex has the same number of neighbors; i.e. every vertex has the same Degree (graph theory), degree or valency. A regular directed graph ...

with 14 edges per vertex. If this graph exists, it cannot have symmetries that take every vertex to every other vertex. Additional restrictions on its possible groups of symmetries are known.

History

The possibility of a graph with these parameters was already suggested in 1969 by Norman L. Biggs, and its existence noted as an open problem by others before Conway. Conway himself had worked on the problem as early as 1975, but offered the prize in 2014 as part of a set of problems posed in the DIMACS Conference on Challenges of Identifying Integer Sequences. Other problems in the set include the thrackle conjecture, the minimum spacing of

Danzer set In geometry, a Danzer set is a set of points that touches every convex body of unit volume. Ludwig Danzer asked whether it is possible for such a set to have bounded density. Several variations of this problem remain unsolved. Formulation A ''D ...

s, and the question of who wins after the move 16 in the game

sylver coinage Sylver coinage is a mathematical game for two players, invented by John H. Conway. The two players take turns naming positive integers that are not the sum of nonnegative multiples of previously named integers. The player who names 1 loses. For i ...

Related graphs

More generally, there are only five possible combinations of parameters for which a strongly regular graph could exist with each edge in a unique triangle and each non-edge forming the diagonal of a unique quadrilateral. It is only known that graphs exist with two of these five combinations. These two graphs are the nine-vertex

Paley graph In mathematics, Paley graphs are undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yiel ...

(the graph of the

3-3 duoprism In the geometry of 4 dimensions, the 3-3 duoprism or triangular duoprism is a four-dimensional convex polytope. Descriptions The duoprism is a 4-polytope that can be constructed using Cartesian product of two polygons. In the case of 3-3 duopr ...

) with parameters (9,4,1,2) and the Berlekamp–van Lint–Seidel graph with parameters (243,22,1,2). The parameters for which graphs are unknown are: (99,14,1,2), (6273,112,1,2) and (494019,994,1,2). The 99-graph problem describes the smallest of these combinations of parameters for which the existence of a graph is unknown.

References

{{reflist, refs= {{citation , last = Biggs , first = Norman , authorlink = Norman L. Biggs , mr = 0327563 , page = 111 , publisher = Cambridge University Press , location = London and New York , series = London Mathematical Society Lecture Note Series , title = Finite Groups of Automorphisms: Course Given at the University of Southampton, October–December 1969 , url = https://books.google.com/books?id=flA4AAAAIAAJ&pg=PA111 , volume = 6 , year = 1971, isbn = 9780521082150 {{citation , last1 = Behbahani , first1 = Majid , last2 = Lam , first2 = Clement , doi = 10.1016/j.disc.2010.10.005 , issue = 2–3 , journal =

Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...

, mr = 2739917 , pages = 132–144 , title = Strongly regular graphs with non-trivial automorphisms , volume = 311 , year = 2011, doi-access = free {{citation , last1 = Brouwer , first1 = A. E. , author1-link = Andries Brouwer , last2 = Neumaier , first2 = A. , doi = 10.1007/BF02122552 , issue = 1 , journal = Combinatorica , mr = 951993 , pages = 57–61 , title = A remark on partial linear spaces of girth 5 with an application to strongly regular graphs , volume = 8 , year = 1988, s2cid = 206812356 , url = https://ir.cwi.nl/pub/1721 {{citation , last = Cameron , first = Peter J. , authorlink = Peter Cameron (mathematician) , isbn = 0-521-45133-7 , mr = 1311922 , page = 331 , publisher = Cambridge University Press , location = Cambridge , title = Combinatorics: topics, techniques, algorithms , url = https://books.google.com/books?id=_aJIKWcifDwC&pg=PA331 , year = 1994 {{citation , last = Conway , first = John H. , author-link = John Horton Conway , accessdate = 2019-02-12 , publisher =

On-Line Encyclopedia of Integer Sequences The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...

, title = Five $1,000 Problems (Update 2017) , url = https://oeis.org/A248380/a248380.pdf. See also {{OEIS el, A248380. {{citation , last = Guy , first = Richard K. , authorlink = Richard K. Guy , editor-last = Kelly , editor-first = L. M. , editor-link = Leroy Milton Kelly , contribution = Problems , doi = 10.1007/BFb0081147 , location = Berlin and New York , mr = 0388240 , pages = 233–244 , publisher = Springer-Verlag , series = Lecture Notes in Mathematics , title = The Geometry of Metric and Linear Spaces , volume = 490 , isbn = 978-3-540-07417-5 , year = 1975. See problem 7 (J. J. Seidel), pp. 237–238. {{citation , last1 = Makhnev , first1 = A. A. , last2 = Minakova , first2 = I. M. , date = January 2004 , doi = 10.1515/156939204872374 , issue = 2 , journal = Discrete Mathematics and Applications , mr = 2069991 , title = On automorphisms of strongly regular graphs with parameters

\lambda=1

\mu=2

, volume = 14, s2cid = 118034273 {{citation , last = Wilbrink , first = H. A. , editor1-last = de Doelder , editor1-first = P. J. , editor2-last = de Graaf , editor2-first = J. , editor3-last = van Lint , editor3-first = J. H. , contribution = On the (99,14,1,2) strongly regular graph , date = August 1984 , pages = 342–355 , publisher = Eindhoven University of Technology , series = EUT Report , title = Papers dedicated to J. J. Seidel , url = https://research.tue.nl/files/2449333/256699.pdf , volume = 84-WSK-03 {{citation , last1 = Zehavi , first1 = Sa'ar , last2 = David de Olivera , first2 = Ivo Fagundes , arxiv = 1707.08047 , title = Not Conway's 99-graph problem , year = 2017 Strongly regular graphs Unsolved problems in graph theory John Horton Conway