Interrelations of
Graph Theory, Combinatorial Designs and the Theory of Error-Correcting Codes, with a point of view from
Applied Algebra, Geometry and Topology.
Symmetry-search endeavor in algebraic and combinatorial structures,
including hypercubes, Cayley graphs
and other colored structures found via codes and designs,
finite fields and geometries.
This has taken us, for example, to the discovery of a vertex-transitive cubic
graph graph on 112 vertices which is not edge-transitive (1993 joint with
A. E. Brouwer and C. Thomassen, previously attributed by I. Bouwer
to R. M. Foster, 1972 unpublished, and described by others since 2005 as
the Ljubljana graph) as a self-complementary graph of the Hamming shell,
namely the complement of the Hamming code in the 7-cube.
At present, involved in geometric and combinatorial properties
of perfect error-correcting codes and related topics in
algebraic graph theory and domination in graphs.
Distribution of Distances in E-sets of Star Graphs
,
preprint,
available as pdf-file.
Great Circle Challenge and Odd Graphs
,
preprint,
available as pdf-file.
(with A. A. Delgado) STS-graphs of perfect codes mod kernel
, Discrete Mathematics 295 (2005), 31-47, available as pdf-file.
(with A. A. Delgado) Classes of Hamilton Cycles in the 5-Cube
, Jour. Combin. Math. Combin. Comput., 61 (2007), 81-95, available as pdf-file.
(NOTE: Reference [3]
at the end of this paper can be found in
http://home.coqui.net/dejterij/mcns/mirame.txt),
as the original site http://www.cnnet.edu became defunct.
On a {K_4,K_{2,2,2}}-ultrahomogeneous graph
,
Australasian Journal of Combinatorics, 44(2009), 63-75.
preprint available as pdf-file,
by courtesy of
http://ajc.maths.uq.edu.au.
(with A. A. Delgado)
Perfect domination in rectangular
grid graphs
, Jour. Combin. Math. Combin. Comput., 70 (2009) 177--196.
available as pdf-file.
A larger version of this manuscript
is available here
(with C. C. Lindner, C. A. Rodger and M. Meszka) Almost resolvable 4-cycle systems
,
Jour. Combin. Math. Combin. Comput., 63 (2007), 173 - 182,
available as pdf-file;
with a Corrigendum/Addendum in Jour. Combin. Math. Combin. Comput., 66 (2008), 297-298, available as
pdf-file, containing a kind contribution by
E. Billington, consisting in using the corrected Example 2.1, contributed by Dejter,
that provided an almost resolvable 4-cycle system of order n = 17, to construct
one of the missing cases, namely the one whose order is n = 33.
(Cases n = 41 and 57 are still not known).
Perfect domination in regular grid graphs,
Australasian Journal of Combinatorics, 92 (2008), 99-114,
available as pdf-file, by courtesy of
http://ajc.maths.uq.edu.au. or in the arXiv with updated references
(M. Mollard, indicated recently that the
1-perfect codes in this paper, appear also in the literature
as perfect codes in Lee metric, starting with the paper by
S. Golomb and K. Welch, Perfect codes in Lee metric
and the packing of polyominoes, SIAM J. Appl. Math.
18 (1970) 302-317, who proved
their existence in all dimensions, as in Theorem 13. Mollard also
mentioned that d-perfect codes (d>1) have sbeen studied in the
2-dimensional grid; their nonexistence has been conjectured for larger
dimensions and proved for d=3 in S. Gravier, M. Mollard and C. Payan,
On the non-existence of 3-dimensional tiling
in the Lee metric, Eur. J. Combin. 19 (1998) 567-572.
By the same authors: Variation on tilings
in Manhattan metric, Geom. Dedicata 76 (1999) 265-273,
On the nonexistence of three-dimensional
tiling in the Lee metric II, Discrete Math., 235 (2001)
151-157. S. Spacapan Non-existence of face-to-face
four dimensional tiling in the Lee metric, European J. Combin. 28 (2007)
127-133, proved the mentioned nonexistence conjecture
for d=4. P. Horak, Tilings in Lee Metric,
Eur. J. Combin., 30 (2009), 480-489, has
a shorter proof of this and extends it for d=5. Also, see Horak's paper
On perfect Lee codes, Discrete Math.,
309 (2009), 5551-5561, where he extends it for d=6).
Quasiperfect domination in triangular
lattices, Discussiones Mathematicae Graph Theory, 29(1) (2009), 179-198,
available here as pdf-file.
SQS-graphs of extended 1-perfect codes,
Congressus Numerantium, 193 (2008), 175-194,
available as pdf-file. A newer version of this publication, SQS-graphs of Solov'eva-Phelps codes, can be found in http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.3178v1.pdf.
On certain $\mathcal C$-ultrahomogeneous graphs obtained from cubic distance transitive graphs codes, preprint,
available as pdf-file.
On a $\vec{C}_4$-ultrahomogeneous oriented graph, preprint,
available as pdf-file.
$\vec{\mathcal C}$-homogeneous graph via ordered pencils, preprint,
available as pdf-file.
