Italo Jose Dejter (December 17, 1939) is an Argentine-born American mathematician, a retired professor of mathematics and computer science from the University of Puerto Rico, (August 1984-February 2018) and a researcher in algebraic topology, differential topology, graph theory, coding theory and combinatorial designs. He obtained a Licentiate degree in mathematics from University of Buenos Aires in 1967, arrived at Rutgers University in 1970 by means of a Guggenheim Fellowship and obtained a Ph.D. degree in mathematics in 1975 under the supervision of Professor Ted Petrie, with support of the National Science Foundation. He was a professor at the Federal University of Santa Catarina, Brazil, from 1977 to 1984, with grants from the National Council for Scientific and Technological Development, (CNPq).
Dejter has been a visiting scholar at a number of research institutions, including University of São Paulo, Instituto Nacional de Matemática Pura e Aplicada, Federal University of Rio Grande do Sul, University of Cambridge, National Autonomous University of Mexico, Simon Fraser University, University of Victoria, New York University, University of Illinois at Urbana–Champaign, McMaster University, DIMACS, Autonomous University of Barcelona, Technical University of Denmark, Auburn University, Polytechnic University of Catalonia, Technical University of Madrid, Charles University, Ottawa University and Simón Bolívar University. The sections below describe the relevance of Dejter's work in the research areas mentioned in the first paragraph above, or in the adjacent box.
Algebraic and differential topology In 1971, Ted PetriePetrie T. "Smooth S1-actions on homotopy complex projective spaces and related topics", Bull. Amer. Math. Soc. 78 (1972) 105–153 conjectured that if X is a closed, smooth 2n-dimensional homotopy complex projective space that admits a nontrivial smooth action of the circle, and if a function h, mapping X onto the 2n-dimensional complex projective space, is a homotopy equivalence, then h preserves the Pontrjagin classes. In 1975, DejterDejter I. J. "Smooth S1-manifolds in the homotopy type of CP3 ", Mich. Math. Jour. 23 (1976), 83–95 proved Petrie's Conjecture for n=3, establishing this way that every closed, smooth, 6-dimensional homotopy complex projective space must be the complex 3-dimensional projective space CP3. Dejter's result is most relevant in view of Petrie's exotic S1-actions on CP3,Petrie T. "Exotic S1-actions on CP3 and related topics", Invent. Math. 17 (1972), 317–327. (apart from the trivial S1-actions on CP3).
Let G be a compact Lie group, let Y be a smooth G-manifold and let F a G-fibre map between G-vector bundles of the same dimension over Y which on each G-fibre is proper and has degree one. Petrie also asked: What are necessary and sufficient conditions for the existence of a smooth G-map properly G-homotopic to F and transverse to the zero-section? DejterDejter I. J. "G-Transversality to CP^n", Springer-Verlag Lecture Notes in Mathematics, 652 (1976), 222–239 provided both types of conditions, which do not close to a necessary and sufficient condition due to a counterexample.
The main tool involved in establishing the results above by reducing differential-topology problems into algebraic-topology solutions is equivariant algebraic K-theory, where equivariance is understood with respect to the group given by the circle, i.e. the unit circle of the complex plane.
Graph theory
Erdős–Pósa theorem and odd cycles In 1962, Paul Erdős and Lajos Pósa proved that for every positive integer k there exists a positive integer k' such that for every graph G, either (i) G has k vertex-disjoint (long and/or even) cycles or (ii) there exists a subset X of less than k' vertices of G such that G \ X has no (long and/or even) cycles. This result, known today as the Erdős–Pósa theorem, cannot be extended to odd cycles. In fact, in 1987 Dejter and Víctor Neumann-LaraDejter I. J.; Neumann-Lara V. "Unboundedness for odd cyclic transversality", Coll. Math. Soc. J. Bolyai, 52 (1987), 195–203 showed that given an integer k > 0, there exists a graph G not possessing disjoint odd cycles such that the number of vertices of G whose removal destroys all odd cycles of G is higher than k.
Ljubljana graph in binary 7-cube In 1993,Brouwer A. E.; Dejter I. J.; Thomassen C. "Highly symmetric subgraphs of hypercubes", J. Algebraic Combinat. 2, 22-25, 1993 Brouwer, Dejter and Thomassen described an undirected, bipartite graph with 112 vertices and 168 edges, (semi-symmetric, that is edge-transitive but not vertex-transitive, cubic graph with diameter 8, radius 7, chromatic number 2, chromatic index 3, girth 10, with exactly 168 cycles of length 10 and 168 cycles of length 12), known since 2002 as the Ljubljana graph. They also established that the Dejter graph,Klin M.; Lauri J.; Ziv-Av M. "Links between two semisymmetric graphs on 112 vertices through the lens of association schemes", Jour. Symbolic Comput., 47–10, 2012, 1175–1191. obtained by deleting a copy of the Hamming code of length 7 from the binary 7-cube, admits a 3-factorization into two copies of the Ljubljana graph. See also.Borges J.; Dejter I. J. "On perfect dominating sets in hypercubes and their complements", J. Combin. Math. Combin. Comput. 20 (1996), 161-173Dejter I. J. "On symmetric subgraphs of the 7-cube: an overview", Discrete Math. 124 (1994) 55–66Dejter I. J. "Symmetry of factors of the 7-cube Hamming shell", J. Combin. Des. 5 (1997), 301–309Dejter I. J.; Guan P. "Square-blocking edge subsets in hypercubes and vertex avoidance", Graph theory, combinatorics, algorithms, and applications (San Francisco, CA, 1989), 162–174, SIAM, Philadelphia, PA, 1991Dejter I. J.; Pujol J. "Perfect domination and symmetry in hypercubes", Proceedings of the Twenty-Sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995). Congr. Numer. 111 (1995), 18–32Dejter I. J.; Weichsel P. M. "Twisted perfect dominating subgraphs of hypercubes", Proceedings of the Twenty-Fourth Southeastern International Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1993). Congr. Numer. 94 (1993), 67–78 Moreover, relations of this subject with square-blocking subsets and with perfect dominating sets (see below) in hypercubes were addressed by Dejter et al. since 1991 in, and .
In fact, two questions were answered in, namely:
(a) How many colors are needed for a coloring of the n-cube without monochromatic 4-cycles or 6-cycles? Brouwer, Dejter and Thomassen showed that 4 colors suffice and thereby settle a problem of Erdős.Erdős P. "Some of my favourite unsolved problems", in: A Tribute to Paul Erdős (A. Baker, B. Bollobás & A. Hajnal, eds.), Cambridge Univ. Press, Cambridge. 1990, 467–478. (Independently found by F.R.K.Chung.Chung F. R. K. "Subgraphs of a hypercube containing no small even cycles", 1. Journal of Graph Theory, 16 (1992) 273–286. Improving on this, Marston ConderConder M. "Hexagon-free subgraphs of hypercubes", Journal of Graph Theory, 17–4 (1993), 477–479. in 1993 showed that for all n not less than 3 the edges of the n-cube can be 3-colored in such a way that there is no monochromatic 4-cycle or 6-cycle).
(b) Which vertex-transitive induced subgraphs does a hypercube have? The Dejter graph mentioned above is 6-regular, vertex-transitive and, as suggested, its edges can be 2-colored so that the two resulting monochromatic subgraphs are isomorphic to the semi-symmetric Ljubljana graph of girth 10.
In 1972, I. Z. BouwerBouwer I. Z. "On edge but not vertex transitive regular graphs", J. Combin. Theory (B) 12 (1972), 32-40. attributed a graph with the mentioned properties of the Ljubljana graph to R. M. Foster.
Coxeter graph and Klein graph In 2012, DejterDejter I. J. From the Coxeter graph to the Klein graph, Journal of Graph Theory, 70-1 (2012), 1–9. showed that the 56-vertex Klein cubic graph F{56}B,Weisstein, Eric W. "Cubic symmetric graph." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CubicSymmetricGraph.html denoted here Γ', can be obtained from the 28-vertex Coxeter cubic graph Γ by zipping adequately the squares of the 24 7-cycles of Γ endowed with an orientation obtained by considering Γ as a -ultrahomogeneousIsaksen D. C.; Jankowski C.; Proctor S. "", Ars Combinatoria, 82 (2007), 83–96. digraph, where is the collection formed both by the oriented 7-cycles and the 2-arcs that tightly fasten those oriented 7-cycles in Γ. In the process, it is seen that Γ' is a C'-ultrahomogeneous (undirected) graph, where C' is the collection formed by both the 7-cycles and the 1-paths that tightly fasten those 7-cycles in Γ'. This yields an embedding of Γ' into a 3-torus T3 which forms the Klein mapSchulte E.; Wills J. M. "A Polyhedral Realization of Felix Klein's Map {3, 7}8 on a Riemann Surface of Genus 3", J. London Math. Soc., s2-32 (1985), 539–547. of Coxeter notation (7,3)8. The dual graph of Γ' in T3 is the distance-regular Klein quartic graph, with corresponding dual map of Coxeter notation (3,7)8. Other aspects of this work are also cited in the following pages:
Bitangents of a quartic.Coxeter graph.Heawood graph.
In 2010, Dejter Dejter I. J. "On a 4-ultrahomogeneous oriented graph", Discrete Mathematics, (2010), 1389–1391. adapted the notion of -ultrahomogeneous graph for digraphs, and presented a strongly connected -ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles with regular indegree and outdegree 3 and no circuits of lengths 2 and 3 by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in which pencils were replaced by ordered pencils.
The study of ultrahomogeneous graphs (respectively, digraphs) can be traced back to Sheehan,Sheehan J. "Smoothly embeddable subgraphs", J. London Math. Soc., s2-9 (1974), 212–218. Gardiner,, Gardiner A. "Homogeneous graphs", Journal of Combinatorial Theory, Series B, 20 (1976), 94–102. Ronse,Ronse C. "On homogeneous graphs", J. London Math. Soc., s2-17 (1978), 375–379. Cameron,Cameron P. J. "6-transitive graphs", J. Combin. Theory Ser. B 28 (1980), 168–179. Gol'fand and Klin,Gol'fand Ja. Ju.; Klin M. H. "On k-homogeneous graphs", Algorithmic studies in combinatorics (Russian), 186 (1978), 76–85. (respectively, Fraïssé,Fraïssé R. "Sur l'extension aux relations de quelques proprietes des ordres", Ann. Sci. Ecole Norm. Sup. 71 (1954), 363–388. Lachlan and Woodrow,A. H. Lachlan A. H.; Woodrow R. "Countable ultrahomogeneous undirected graphs", Trans. Amer. Math. Soc. 262 (1980), 51–94. CherlinCherlin G. L. "The classification of countable homogeneous directed graphs and countable homogeneous n-tournaments", Memoirs Amer. Math. Soc., vol. 131, number 612, Providence RI, January 1988.). See also page 77 in Bondy and Murty.Bondy A.; Murty U.S.R.; Graph Theory, Springer-Verlag, 2008.
Kd-ultrahomogeneous configurations
Motivated in 2013Dejter I. J. "On a K4-UH self-dual 1-configuration (10241, J. Combin. Math. Combi. Comput. 95 (2015), 127-146 (arXiv:1002:0588) [math.CO]. by the study of connected Menger graphs Coxeter H. S. M. "Self-dual configurations and regular graphs", Bull. Amer. Math. Soc., 56 (1950), 413-455; http://www.ams.org/journals/bull/1950-56-05/S0002-9904-1950-09407-5/S0002-9904-1950-09407-5.pdf. of self-dual 1-configurations (nd)1 Colbourn C. J.; Dinitz J. H. "The CRC Handbook of Combinatorial Designs", CRC, 1996. expressible as Kd-ultrahomogeneous graphs, Dejter wondered for which values of n such graphs exist, as they would yield the most symmetrical, connected, edge-disjoint unions of n copies of Kd on n vertices in which the roles of vertices and copies of Kd are interchangeable. For d=4, known values of n are: n=13, 21Grünbaum B. "Configurations of Points and Lines", Grad. Texts