Subadditive set function

From formulasearchengine
Revision as of 16:49, 22 October 2013 by en>Wdvorak (→‎Examples of subadditive functions)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

Template:Infobox scientist

Italo Jose Dejter is an Argentine-born American mathematician and a professor at the University of Puerto Rico (UPRRP)[1] since 1984. His research interests are in algebraic topology, differential topology, graph theory, coding theory and design theory.

In 1967, Dejter received a Licentiate degree in Mathematics from the University of Buenos Aires, and in 1975, he received a Ph.D. degree in Mathematics from Rutgers University under the supervision of Ted Petrie.[2] From 1977 to 1984, Dejter was a professor at Universidade Federal de Santa Catarina. Since 1993,[3] he possesses an Erdős number of 2.[4]

Dejter has been a visiting scholar at University of São Paulo, Instituto Nacional de Matemática Pura e Aplicada, Universidade Federal do 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, Simón Bolívar University, etc. The sections below describe the relevance of Dejter's work in the research areas mentioned in the first paragraph above, or on the box to the right.

Algebraic and differential topology

In 1971, Ted Petrie[5] 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, Dejter[6] 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,[7] (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[5] 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? Dejter[8] provided both types of conditions, which do not close to a necessary and sufficient condition due to a counterexample.[8]

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 Neumann-Lara [9] 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,[3] 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[3] also established that the Dejter graph,[10] 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.[11][12][13][14][15][16] 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 ,[14][15][16] and .[11]

In fact, two questions were answered in,[3] 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[3] showed that 4 colors suffice and thereby settle a problem of Erdős.[17] (Independently found by F.R.K.Chung.[18] Improving on this, Marston Conder[19] 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. Bouwer[20] attributed a graph with the mentioned properties of the Ljubljana graph to R. M. Foster.

Coxeter graph and Klein graph

In 2012, Dejter[21] showed that the 56-vertex Klein cubic graph F{56}B, [22] 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 C-ultrahomogeneous[23] digraph, where C 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 map[24] 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 [25] adapted the notion of C-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,[26] Gardiner,[27] Ronse,[28] Cameron,[29] Gol'fand and Klin,[30] (respectively, Fraïssé,[31] Lachlan and Woodrow,[32] Cherlin[33]). See also page 77 in Bondy and Murty.[34]

Kd-ultrahomogeneous configurations

Motivated in 2013[35] by the study of connected Menger graphs [36] of self-dual 1-configurations (nd)1 [37] [38] expressible as Kd-ultrahomogeneous graphs, Dejter found it interesting the question of for which values of n such graphs exist, because 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, 21[39][40][41] and n=42,[42] (this last reference, by Dejter in 2009, yielding a graph G for which each isomorphism between two of the 42 copies of K4 or two of the 21 copies of K2,2,2 in G extends to an automorphism of G). While it would be of interest to determine the spectrum and multiplicities of the involved values of n, Dejter[35] contributes the value of n=102 via the Biggs-Smith association scheme (presented via sextets[43] mod 17), shown to control attachment of 102 (cuboctahedral) copies of the line graph of the 3-cube to the 102 (tetrahedral) copies of K4, these sharing each triangle with two copies of the cuboctahedral copies and guaranteeing that the distance 3-graph of the Biggs-Smith graph is the Menger graph of a self-dual 1-configuration (1024)1. This result[35] was obtained as an application of a transformation of distance-transitive graphs into C-UH graphs that yielded the above mentioned paper[21] and also allowed to confront ,[44] as digraphs, the Pappus graph to the Desargues graph.

These applications as well as [45] use the following definition. Given a family C of digraphs, a digraph G is said to be C-ultrahomogeneous if every isomorphism between two induced members of C in G extends to an automorphism of G. In,[45] it is shown that exactly 7 distance-transitive cubic graphs among the existing 12 possess a particular ultrahomogeneous property with respect to oriented cycles realizing the girth that allows the construction of a related Cayley digraph with similar ultrahomogeneous properties in which those oriented cycles appear minimally “pulled apart”, or “separated” and whose description is truly beautiful and insightful.

Hamiltonicity in graphs

In 1983, Dejter[46] found that an equivalent condition for the existence of a Z4-Hamilton cycle on the graph of chessknight moves of the usual type (1,2),(resp (1,4)) on the 2nx2n-board is that n is odd larger than 2, (resp. 4). These results are cited by I. Parberry[47],[48] in relation to the algorithmic aspects of the knight's tour problem.

In 1985, Dejter[49] provided a sufficient condition for the existence of Hamilton cycles in the middle-levels graph, whose existence was conjectured by I. Havel in 1983.[50] This condition was used to produce these cycles in a number of papers by Dejter and former students.[51][52][53][54][55][56] In 2013, Dejter[57] returned to this problem and proposed a canonical ordering of the vertices of the reduced graph of each middle levels graph obtained via a Catalan numeral system and Kierstead-Trotter lexical matching colors,[58] which allows for a search of extremal Hamilton cycles both in the Catalan numeral ordering and in the matching coloring ordering.

In 1988, Dejter[59] showed that for any positive integer n, all 2-covering graphs of the complete graph Kn on n vertices can be determined; in addition, he showed that among them there is only one graph that is connected and with maximal automorphism group, which happens to be bipartite; Dejter also showed that an i-covering graph of Kn is hamiltonian, for i less than 4, and that properly minimal connected non-hamiltonian covering graphs of Kn are obtained which are 4-coverings of Kn; also, non-hamiltonian connected 6-coverings of Kn were constructed in that work.

In 1990, Dejter[60] showed that if n and r are integers larger than 0 with n+r larger than 2, then the difference of two concentric square boards A and B with (n + 2r)2 and n2 entries respectively has a chessknight Hamilton cycle invariant under quarter-turns if and only if r is larger than 2 and either n or r is odd.

In 1991, Dejter and Neumann-Lara [61] showed that given a group Zn acting freely on a graph G, the notion of a voltage graph[62] is applied to the search for Hamilton cycles in G invariant under an action of Zn on G. As an application, for n = 2 and 4, equivalent conditions and lower bounds for chessknight Hamilton cycles containing paths spanning square quadrants and rectangular half-boards were found, respectively.

Perfect dominating sets

A perfect dominating set S of a graph G is a set of vertices of G such that every vertex of G is either in S or is adjacent to exactly one vertex of S. Weichsel[63] showed that a perfect dominating set of the n-cube Qn induces a subgraph of Qn whose components are isomorphic to hypercubes and conjectured that each of these hypercubes has the same dimension. In 1993, Dejter and Weichsel[16] presented the first known cases in which those components have the same dimension but different directions, namely in the 8-cube with components that are 1-cubes formed each by one edge, with the involved edges happening in:

(a) 4 different directions, as told by Alexander Felzenbaum to Weichsel in Rehovot, Israel, 1988;

(b) 8 different directions, which involves the Hamming code of length 7, the Heawood graph, the Fano plane and the Steiner triple system of order 7.

The result of (a) above is immediately extended to perfect dominating sets in cubes of dimensions which are powers of 2 whose components contain each an only edge in half the coordinate direction. On the other hand in 1991, Dejter and Phelps[64] extended the result of (b) above again to cubes whose dimensions are powers of 2, with components composed each by a unique edge in all coordinate directions. (However, this result is not yet extended to q-ary cubes, as planned by the authors).

The Weichsel conjecture[63] was answered in the affirmative by Östergård and Weakley,[65] who found a perfect dominating set in the 13-cube whose components are 26 4-cubes and 288 isolated vertices. Dejter and Phelps[66] gave a short and elegant proof of this result that uses Östergård's email comment on the fact that the 288 isolated vertices above correspond to the nonzero codewords of the ternary Hamming code of length 13.

Efficient dominating sets

An E-chain is a countable family of nested graphs, each of which has an efficient dominating set. The Hamming codes in the n-cubes provide a classical example of E-chains. Dejter and Serra[67] gave a constructing tool to produce E-chains of Cayley graphs. This tool was used to construct infinite families of E-chains of Cayley graphs on symmetric groups. These families include the well-known star graphs,[68] for which the efficient domination property was proved by Arumugam and Kala,[69] and pancake graphs.[68] Given a tree T, the T-graph associated to T seems to be a natural candidate of a graph with an efficient dominating set. However, Dejter and Serra proved that a T-graph has an efficient dominating set if and only if T is a star. Further study on threaded distance trees and E-sets of star graphs was conducted by Dejter.[70] In 2012, Dejter adapted the results in this paragraph to the case of digraphs.[71] In fact, worst-case efficient dominating sets in digraphs are conceived so that their presence in certain strong digraphs corresponds to that of efficient dominating sets in star graphs. The fact that the star graphs form a so-called dense segmental neighborly E-chain[67] is reflected in a corresponding fact for digraphs.

Quasiperfect dominating sets

In 2009,[72] Dejter defined a vertex subset S of a graph G as a quasiperfect dominating set in G if each vertex v of G not in S is adjacent to dv ∈{1,2} vertices of S, and then investigated perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol {3,6} and in its toroidal quotient graphs, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets S with induced components of the form Kν, where ν ∈{1,2,3} depends only on S.

Coding theory

Invariants of perfect error-correcting codes

Invariants of perfect error-correcting codes were addressed by Dejter in,[73][74] and Dejter and Delgado[75] in which it is shown that a perfect 1-error-correcting code C is ‘foldable’ over its kernel via the Steiner triple systems associated to its codewords. The resulting ‘folding’ produces a graph invariant for C via Pasch configurations and tensors. Moreover, the invariant is complete for Vasil’ev codes[76] of length 15 as viewed by F. Hergert,[77] showing the existence of nonadditive propelinear 1-perfect codes,[78][79] and allowing to visualize a propelinear code by means of the commutative group formed by its classes mod kernel, as well as to generalize the notion of a propelinear code by extending the involved composition of permutations to a more general group product.

Generalizing perfect Lee codes

Motivated by an application problem in computer architecture, Araujo, Dejter and Horak[80] introduced a notion of perfect distance-dominating set, PDDS, in a graph, constituting a generalization of perfect Lee codes,[81] diameter perfect codes,[82] and other codes and dominating sets, and thus initiating a systematic study of such vertex sets. Some of these sets, related to the motivating application, were constructed, and the non-existence of others was demonstrated. In fact, an extension of the long-standing Golomb-Welch conjecture,[81] in terms of PDDSs, was stated.

Total perfect codes

According to Dejter and Delgado,[83] given a vertex subset S' of a side Pm of an m x n grid graph G, the perfect dominating sets S in G with S' being the intersection of S with V(Pm) can be determined via an exhaustive algorithm of running time O(2m+n). Extending the algorithm to infinite-grid graphs of width m-1, periodicity makes the binary decision tree prunable into a finite threaded tree, a closed walk of which yields all such sets S. The graphs induced by the complements of such sets S can be codified by arrays of ordered pairs of positive integers, for the growth and determination of which a speedier algorithm exists. A recent characterization of grid graphs having total perfect codes S (i.e. with just 1-cubes as induced components, also called 1-PDDS[80] and DPL(2,4)[82]), due to Klostermeyer and Goldwasser,[84] allowed Dejter and Delgado[83] to show that these sets S are restrictions of only one total perfect code S1 in the planar integer lattice graph, with the extra-bonus that the complement of S1 yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in the planar integer lattice graph are in 1-1 correspondence with the doubly infinite {0,1}-sequences.

Dejter showed[85] that there is an uncountable number of parallel total perfect codes in the planar integer lattice graph L but that in contrast, there is just one 1-perfect code and just one total perfect code in L, this code restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane); in particular, Dejter characterized all cycle products Cm x Cn containing parallel total perfect codes, and the d-perfect and total perfect code partitions of L and Cm x Cn, the former having as quotient graph the undirected Cayley graphs of the cyclic group of order 2d2+2d+1 with generator set {1,2d2}.

In 2012, Araujo and Dejter[86] made a conjecturing contribution to the classification of lattice-like total perfect codes in n-dimensional integer lattices via pairs (G,F) formed by abelian groups G and homomorphisms F from Zn onto G, in the line of .[80]

Design theory

Since 1994, Dejter intervened in several projects in Design Theory initially suggested by A. Rosa, C. C. Lindner and C. A. Rodger and also worked upon with E. Mendelsohn, F. Franek, D. Pike, P. A. Adams, E. J. Billington, D. G. Hoffman, M. Meszka and others, which produced results in the following subjects:

Invariants for 2-factorization and cycle systems,[87]

Triangles in 2-factorizations,[88][89]

Number of 4-cycles in 2-factorizations of complete graphs,[90]

Directed almost resolvable Hamilton-Waterloo problem,[91]

Number of 4-cycles in 2-factorizations of K2n} minus a 1-factor,[92]

Almost 4-cycle systems,[93]

Critical sets for the completion of Latin squares[94]

Almost resolvable maximum packings of complete graphs with 4-cycles.[95]

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Template:Persondata

  1. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named UPRRP
  2. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named gen
  3. ↑ 3.0 3.1 3.2 3.3 3.4 Brouwer A. E.; Dejter I. J.; Thomassen C. "Highly symmetric subgraphs of hypercubes", J. Algebraic Combinat. 2, 22-25, 1993
  4. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named erdos
  5. ↑ 5.0 5.1 Petrie T. "Smooth S1-actions on homotopy complex projective spaces and related topics", Bull. Amer. Math. Soc. 78 (1972) 105–153
  6. ↑ Dejter I. J. "Smooth S1-manifolds in the homotopy type of CP3 ", Mich. Math. Jour. 23 (1976), 83–95
  7. ↑ Petrie T. "Exotic S1-actions on CP3 and related topics", Invent. Math. 17 (1972), 317–327.
  8. ↑ 8.0 8.1 Dejter I. J. "G-Transversality to CP^n", Springer-Verlag Lecture Notes in Mathematics, 652 (1976), 222–239
  9. ↑ Dejter I. J.; Neumann-Lara V. "Unboundedness for odd cyclic transversality", Coll. Math. Soc. J. Bolyai, 52 (1987), 195–203
  10. ↑ 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.
  11. ↑ 11.0 11.1 Borges J.; Dejter I. J. "On perfect dominating sets in hypercubes and their complements", J. Combin. Math. Combin. Comput. 20 (1996), 161-173
  12. ↑ Dejter I. J. "On symmetric subgraphs of the 7-cube: an overview", Discrete Math. 124 (1994) 55–66
  13. ↑ Dejter I. J. "Symmetry of factors of the 7-cube Hamming shell", J. Combin. Des. 5 (1997), 301–309
  14. ↑ 14.0 14.1 Dejter 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, 1991
  15. ↑ 15.0 15.1 Dejter 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), 18–32
  16. ↑ 16.0 16.1 16.2 Dejter 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
  17. ↑ 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.
  18. ↑ Chung F. R. K. "Subgraphs of a hypercube containing no small even cycles", 1. Journal of Graph Theory, 16 (1992) 273–286.
  19. ↑ Conder M. "Hexagon-free subgraphs of hypercubes", Journal of Graph Theory, 17–4 (1993), 477–479.
  20. ↑ Bouwer I. Z. "On edge but not vertex transitive regular graphs", J. Combin. Theory (B) 12 (1972), 32-40.
  21. ↑ 21.0 21.1 Dejter I. J. From the Coxeter graph to the Klein graph, Journal of Graph Theory, 70-1 (2012), 1–9.
  22. ↑ Weisstein, Eric W. "Cubic symmetric graph." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/CubicSymmetricGraph.html
  23. ↑ Isaksen D. C.; Jankowski C.; Proctor S. "On K,sub*-ultrahomogeneous graphs", Ars Combinatoria, 82 (2007), 83–96.
  24. ↑ Schulte 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.
  25. ↑ Dejter I. J. "On a -ultrahomogeneous oriented graph", Discrete Mathematics, (2010), 1389–1391.
  26. ↑ Sheehan J. "Smoothly embeddable subgraphs", J. London Math. Soc., s2-9 (1974), 212–218.
  27. ↑ , Gardiner A. "Homogeneous graphs", J. Combinatorial Theory (B), 20 (1976), 94–102.
  28. ↑ Ronse C. "On homogeneous graphs", J. London Math. Soc., s2-17 (1978), 375–379.
  29. ↑ Cameron P. J. "6-transitive graphs", J. Combin. Theory Ser. B 28 (1980), 168–179.
  30. ↑ Gol'fand Ja. Ju.; Klin M. H. "On k-homogeneous graphs", Algorithmic studies in combinatorics (Russian), 186 (1978), 76–85.
  31. ↑ Fraïssé R. "Sur l'extension aux relations de quelques proprietes des ordres", Ann. Sci. Ecole Norm. Sup. 71 (1954), 363–388.
  32. ↑ A. H. Lachlan A. H.; Woodrow R. "Countable ultrahomogeneous undirected graphs", Trans. Amer. Math. Soc. 262 (1980), 51–94.
  33. ↑ Cherlin 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.
  34. ↑ Bondy A.; Murty U.S.R.; Graph Theory, Springer-Verlag, 2008.
  35. ↑ 35.0 35.1 35.2 Dejter I. J. "On a K4-UH self-dual 1-configuration (10241</math>, arXiv:1002.0588 [math.CO]</a>.
  36. ↑ 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.
  37. ↑ Gropp , "On symmetrical spatial configurations", Discrete Math., 125 (1994), 201-209; http://ac.els-cdn.com/0012365X94901619/1-s2.0-0012365X94901619-main.pdf?_tid=2ebc327c-5d66-11e2-9bc9-00000aab0f6c&acdnat=1358070604_ce3ded8f20a6d89573fbec58318de118.
  38. ↑ Colbourn C. J.; Dinitz J. H. "The CRC Handbook of Combinatorial Designs", CRC, 1996.
  39. ↑ Grünbaum B. "Configurations of Points and Lines", Grad. Texts in Math. 103, Amer. Math. Soc, Providence R.I., 2009.
  40. ↑ Grünbaum B.; Rigby J. F. "The real configuration (214)", Jour. London Math. Soc., Sec. Ser. 41(2) (1990), 336–346.
  41. ↑ Pisanski T.; Servatius B. "Configurations from a Graphical Viewpoint", Birkhäuser, 2013.
  42. ↑ Dejter I. J. "On a {K4,K2,2,2}-ultrahomogeneous graph", Australasian Journal of Combinatorics, 44 (2009), 63-76.
  43. ↑ Biggs N. L.; Hoare M. J. "The sextet construction for cubic graphs", Combinatorica, 3 (1983), 153-165.
  44. ↑ Dejter I. J. "Pappus-Desargues digraph confrontation", Jour. Combin. Math. Combin. Comput", to appear 2013, arXiv:0904.1096 [math.CO]
  45. ↑ 45.0 45.1 Dejter I. J. "Orienting and separating distance-transitive graphs", Ars Mathematica Contemporanea, 5 (2012) 221-236
  46. ↑ I. J. Dejter "Equivalent conditions for Euler problem on Z4-Hamilton cycles", Ars Combinatoria, 16B, (1983), 285-295;http://home.coqui.net/dejterij/arsc1983.pdf.
  47. ↑ https://larc.unt.edu/ian/research/puzzles/knightstour/
  48. ↑ I. Parberry "An efficient algorithm for the Knight�s tour problem", Discrete Applied Mathematics, 73, (1997), 251-260
  49. ↑ Dejter I. J. "Hamilton cycles and quotients of bipartite graphs", in Y. Alavi et al., eds., Graph Theory and its Appl. to Alg. and Comp. Sci.}, Wyley, 1985, 189-199;http://home.coqui.net/dejterij/dejflori.pdf.
  50. ↑ Havel I. "Semipaths in directed cubes", in: M. Fiedler (Ed.), Graphs and other Combinatorial Topics, Teubner-Texte Math., Teubner, Leipzig, 1983, pp. 101-108.
  51. ↑ Dejter I. J. "Stratification for hamiltonicity", Congressus Numeranium, 47 (1985) 265-272.
  52. ↑ Dejter I. J.; Quintana J. "Long cycles in revolving door graphs", Congressus Numerantium, 60 (1987), 163-168.
  53. ↑ Dejter I. J.; Cordova J,; Quintana J. "Two Hamilton cycles in bipartite reflective Kneser graphs", Discrete Math. 72 (1988) 63-70.
  54. ↑ Dejter I. J.; Quintana J. "On an extension of a conjecture of I. Havel", in Y. Alavi et al. eds., Graph Theory, Combin. and Appl., J. Wiley 1991, vol I, 327-342.
  55. ↑ Dejter I. J.; Cedeno W.; Jauregui V. "Frucht diagrams, Boolean graphs and Hamilton cycles", Scientia, Ser. A, Math. Sci., 5 (1992/93) 21-37.
  56. ↑ Dejter I. J.; Cedeno W.; Jauregui V. "A note on F-diagrams, Boolean graphs and Hamilton cycles", Discrete Math. 114 (1993), 131-135.
  57. ↑ Dejter I. J. "Catalan structure of middle-levels graphs", http://home.coqui.net/dejterij/mofongo.pdf.
  58. ↑ Kierstead H. A.; Trotter W. T. "Explicit matchings in the middle two levels of the boolean algebra", Order 5 (1988), 163-171.
  59. ↑ I. J. Dejter "Minimal hamiltonian and nonhamiltonian covering graphs of Kn", Ars Combinatoria, 25-C, (1988), 63-71;http://home.coqui.net/dejterij/arsc1988.pdf
  60. ↑ I. J. Dejter "Quarter-turns and Hamilton cycles for annular chessknight graphs", Scientia, Ser. A, Math. Sci., 4 (1990/91), 21-29;http://home.coqui.net/dejterij/knight.pdf.
  61. ↑ I. J. Dejter and V. Neumann-Lara "Voltage graphs and Hamilton cycles", in V. Kulli ed., Advances in Graph Theory, Vishwa Int. Publ., Gulbarga, India, 1991, 141-153.
  62. ↑ J.L. Gross and T.W. Tucker "Topological Graph Theory" Wiley, New York (1987).
  63. ↑ 63.0 63.1 Weichsel P. "Dominating sets in n-cubes", Jour. of Graph Theory, 18 (1994), 479-488
  64. ↑ Dejter. I. J.; Phelps K. T. "On perfect domination of q-ary cubes", preprint, http://home.coqui.net/dejterij/terfallo.pdf
  65. ↑ Östergård P.; Weakley W. D. "Constructing covering codes with given automorphisms", Des. Codes Cryptogr. 16 (1999), 65-73
  66. ↑ Dejter I. J.; Phelps K. T. "Ternary Hamming and Binary Perfect Covering Codes", in: A. Barg and S. Litsyn, eds., Codes and Association Schemes, DIMACS Ser. Discrete Math. Theoret. Comput Sci. 56, Amer. Math. Soc., Providence, RI, 111--113"
  67. ↑ 67.0 67.1 Dejter I. J.; Serra O. "Efficient dominating sets in Cayley graphs" , Discrete Appl. Math., 129 (2003), no. 2-3, 319-328.
  68. ↑ 68.0 68.1 Akers S.B.; Krishnamurthy B. "A group theoretic model for symmetric interconnection networks", IEEE Trans. Comput., 38 (1989), 555-565.
  69. ↑ Arumugam S.; Kala R. "Domination Parameters of Star Graphs", Ars Combinatoria, 44 (1996) 93-96
  70. ↑ Dejter I. J. "Star graphs: threaded distance trees and E-sets", J. Combin. Math. Combin. Comput. 77 (2011), 3-16.
  71. ↑ Dejter I. J. "Worst-case efficient dominating sets in digraphs", Discrete Applied Mathematics, 161 (2013) 944–952. First Online DOI 10.1016/j.dam.2012.11.016
  72. ↑ Dejter I. J. "Quasiperfect domination in triangular lattices" Discussiones Mathematicae Graph Theory, 29(1) (2009), 179-198.
  73. ↑ Dejter I. J. "SQS-graphs of extended 1-perfect codes", Congressus Numerantium, 193 (2008), 175-194.
  74. ↑ Dejter I. J. "STS-Graphical invariant for perfect codes", J. Combin. Math. Combin. Comput., 36 (2001), 65-82.
  75. ↑ Dejter I. J.; Delgado A. A. "STS-Graphs of perfect codes mod kernel", Discrete Mathematics, 253 (2005), 31-47.
  76. ↑ Vasil’ev Y. L. "On nongroup close-packed codes", Problem of Cybernetics, 8 (1962) 375-378 (in Russian).
  77. ↑ Hergert F, "The equivalence classes of the Vasil’ev codes of length 15", Springer-Verlag Lecture Notes 969 (1982) 176-186.
  78. ↑ Rifà J.; Basart J. M.; Huguet L. "On completely regular propelinear codes" AAECC (1988) 341-355
  79. ↑ Rifà J.; Pujol J. "Translation invariant propelinear codes, Transact. Info. Th., IEEE, 43(1997) 590-598.
  80. ↑ 80.0 80.1 80.2 Araujo C; Dejter I. J.; Horak P. "generalization of Lee codes", Designs, Codes and Cryptography, Online First, 2012, DOI: 10.1007/s10623-012-9666-6;http://home.coqui.net/dejterij/adp-yo.pdf.
  81. ↑ 81.0 81.1 Golomb S. W.; Welsh L. R. "Perfect codes in the Lee metric and the packing of polyominoes", SIAM J. Applied Math. 18 (1970), 302-317.
  82. ↑ 82.0 82.1 Horak, P.; AlBdaiwi, B.F "Diameter Perfect Lee Codes", IEEE Transactions on Information Theory 58-8 (2012), 5490-5499.
  83. ↑ 83.0 83.1 Dejter I. J.; Delgado A. A. "Perfect domination in rectangular grid graphs", J. Combin. Math. Combin. Comput., 70 (2009), 177-196.
  84. ↑ Klostermeyer W. F.; Goldwasser J. L. "Total Perfect Codes in Grid Codes", Bull. Inst. Comb. Appl., 46(2006) 61-68.
  85. ↑ Dejter I. J. "Perfect domination in regular grid graphs", Austral. Jour. Combin., 42 (2008), 99-114
  86. ↑ Dejter I. J.; Araujo C. "Lattice-like total perfect codes", Discussiones Mathematicae Graph Theory, 34 (2014) 57–74, doi:10.7151/dmgt.1715, http://home.coqui.net/dejterij/aug05a.pdf
  87. ↑ Dejter I. J.; Rivera-Vega P. I.; Rosa A. "Invariants for 2-factorizations and cycle systems", J. Combin. Math. Combin. Comput., 16 (1994), 129-152.
  88. ↑ Dejter I. J.; Franek F.; Mendelsohn E.; Rosa A. "Triangles in 2-factorizations", Journal of Graph Theory, 26 (1997) 83-94.
  89. ↑ Dejter I. J.; Franek F.; Rosa A. "A Completion conjecture for Kirkman triple systems", Utilitas Mathematica, 50 (1996) 97-102
  90. ↑ Dejter I. J.; Lindner C. C.; Rosa A. "The number of 4-cycles in 2-factorizations of Kn", J. Combin. Math. Combin. Comput., 28 (1998), 101-112.
  91. ↑ Dejter I. J.; Pike D.; Rodger C. A. "The directed almost resolvable Hamilton-Waterloo problem", Australas. J. Combin., 18 (1998), 201-208.
  92. ↑ Adams P. A., Billington E. J.; Lindner C. C. "The number of 4-cycles in 2-factorizations of K2n minus a 1-factor}, Discrete Math., 220 (2000), no.1-3, 1-11.
  93. ↑ Dejter I. J.; Lindner C. C.; Rodger C. A.; Meszka M. "Almost resolvable 4-cycle systems", J. Combin. Math. Combin. Comput., 63 (2007), 173-182.
  94. ↑ Horak P.; Dejter I. J. "Completing Latin squares: critical sets, II", Jour. Combin. Des., 15 (2007), 177-83.
  95. ↑ Billington E. J.; Dejter I. J.; Hoffman D. G.; Lindner C. C. "Almost resolvable maximum packings of complete graphs with 4-cycles", Graphs and Combinatorics, 27 (2011), no. 2, 161-170
Retrieved from "https://en.formulasearchengine.com/index.php?title=Subadditive_set_function&oldid=28348"