arXiv subjects

Italo J. Dejter

Publications and source records attributed to Italo J. Dejter.

33 records · Page 2Linked to original sources

arXiv2014

On rainbow tetrahedra in Cayley graphs

Let $Γ_n$ be the complete undirected Cayley graph of the odd cyclic group $Z_n$. Connected graphs whose vertices are rainbow tetrahedra in $Γ_n$ are studied, with any two such vertices adjacent if and only if they share (as tetrahedra) precisely two distinct triangles. This yields graphs $G$ of largest degree 6, asymptotic diameter $|V(G)|^{1/3}$ and almost all vertices with degree: {\bf(a)} 6 in $G$; {\bf(b)} 4 in exactly six connected subgraphs of the $(3,6,3,6)$-semi-regular tessellation; and {\bf(c)} 3 in exactly four connected subgraphs of the $\{6,3\}$-regular hexagonal tessellation. These vertices have as closed neighborhoods the union (in a fixed way) of closed neighborhoods in the ten respective resulting tessellations. Generalizing asymptotic results are discussed as well.

math.CO↗

arXiv2013

A Generalization of Lee Codes

Motivated by a problem in computer architecture we introduce a notion of the perfect distance-dominating set, PDDS, in a graph. PDDSs constitute a generalization of perfect Lee codes, diameter perfect codes, as well as other codes and dominating sets. In this paper we initiate a systematic study of PDDSs. PDDSs related to the application will be constructed and the non-existence of some PDDSs will be shown. In addition, an extension of the long-standing Golomb-Welch conjecture, in terms of PDDS, will be stated. We note that all constructed PDDSs are lattice-like which is a very important feature from the practical point of view as in this case decoding algorithms tend to be much simpler.

cs.DM↗

arXiv2013

On a $K_4$-UH self-dual 1-configuration $(102_4)_1$

Self-dual 1-configurations $(n_d)_1$ possess their Menger graph $\mathcal Y$ most $K_4$-separated among connected self-dual configurations $(n_d)$. Such $\mathcal Y$ is most symmetric if $K_d$-ultrahomogeneous. In this work, such a $\mathcal Y$ is presented for $(n,d)=(102,4)$ and shown to relate $n$ copies of the cuboctahedral graph $L(Q_3)$ to the $n$ copies of $K_d$; these are shown to share each copy of $K_3$ exactly with two copies of $L(Q_3)$.

math.CO↗

arXiv2012

Worst-case efficient dominating sets in digraphs

Let $1\le n\in\Z$. {\it Worst-case efficient dominating sets in digraphs} are conceived so that their presence in certain strong digraphs $\vec{ST}_n$ corresponds to that of efficient dominating sets in star graphs $ST_n$: The fact that the star graphs $ST_n$ form a so-called dense segmental neighborly E-chain is reflected in a corresponding fact for the digraphs $\vec{ST}_n$. Related chains of graphs and open problems are presented as well.

math.CO↗

arXiv2012

Pappus-Desargues digraph confrontation

Like the Coxeter graph became reattached into the Klein graph in [2], the Levi graphs of the $9_3$ and $10_3$ self-dual configurations, known as the Pappus and Desargues ($k$-transitive) graphs $\mathcal P$ and $\mathcal D$ (where $k=3$), also admit reattachments of the distance-$(k-1)$ graphs of half of their oriented shortest cycles via orientation assignments on their common $(k-1)$-arcs, concurrent for ${\mathcal P}$ and opposite for $\mathcal D$, now into 2 disjoint copies of their corresponding Menger graphs. Here, $\mathcal P$ is the unique cubic distance-transitive (or CDT) graph with the concurrent-reattachment behavior while $\mathcal D$ is one of 7 CDT graphs with the opposite-reattachment behavior, that include the Coxeter graph. Thus, $\mathcal P$ and $\mathcal D$ confront each other in these respects, obtained via $\mathcal C$-ultrahomogeneous graph techniques [3,4] that allow to characterize the obtained reattachment Menger graphs in the same terms.

math.CO↗

arXiv2012

From the Coxeter graph to the Klein graph

We show that the 56-vertex Klein cubic graph $\G'$ can be obtained from the 28-vertex Coxeter cubic graph $\G$ by 'zipping' adequately the squares of the 24 7-cycles of $\G$ endowed with an orientation obtained by considering $\G$ as a $\mathcal C$-ultrahomogeneous digraph, where $\mathcal C$ is the collection formed by both the oriented 7-cycles $\vec{C}_7$ and the 2-arcs $\vec{P}_3$ that tightly fasten those $\vec{C}_7$ in $\G$. In the process, it is seen that $\G'$ is a ${\mathcal C}'$-ultrahomogeneous (undirected) graph, where ${\mathcal C}'$ is the collection formed by both the 7-cycles $C_7$ and the 1-paths $P_2$ that tightly fasten those $C_7$ in $\G'$. This yields an embedding of $\G'$ into a 3-torus $T_3$ which forms the Klein map of Coxeter notation $(7,3)_8$. The dual graph of $\G'$ in $T_3$ is the distance-regular Klein quartic graph, with corresponding dual map of Coxeter notation $(3,7)_8$.

math.CO↗

arXiv2012

Orienting and separating distance-transitive graphs

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. This work is proposed as the initiation of a study of similar ultrahomogeneous properties for distance-transitive graphs in general with the aim of generalizing to constructions of similar related "separator" Cayley digraphs.

math.CO↗

arXiv2010

Star graphs: threaded distance trees and E-sets

The distribution of distances in the star graph $ST_n$, ($1<n\in\Z$), is established, and subsequently a threaded binary tree is obtained that realizes an orientation of $ST_n$ whose levels are given by the distances to the identity permutation, via a pruning algorithm followed by a threading algorithm. In the process, the distributions of distances of the efficient dominating sets of $ST_n$ are determined.

math.CO↗

arXiv2010

Perfect domination in regular grid graphs

We show there is an uncountable number of parallel total perfect codes in the integer lattice graph $Λ$ of $\R^2$. In contrast, there is just one 1-perfect code in $Λ$ and one total perfect code in $Λ$ restricting to total perfect codes of rectangular grid graphs (yielding an asymmetric, Penrose, tiling of the plane). We characterize all cycle products $C_m\times C_n$ with parallel total perfect codes, and the $d$-perfect and total perfect code partitions of $Λ$ and $C_m\times C_n$, the former having as quotient graph the undirected Cayley graphs of $\Z_{2d^2+2d+1}$ with generator set $\{1,2d^2\}$. For $r>1$, generalization for 1-perfect codes is provided in the integer lattice of $\R^r$ and in the products of $r$ cycles, with partition quotient graph $K_{2r+1}$ taken as the undirected Cayley graph of $\Z_{2r+1}$ with generator set $\{1,...,r\}$.

math.CO↗

arXiv2010

SQS-graphs of extended 1-perfect codes

A binary extended 1-perfect code $\mathcal C$ folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for $\mathcal C$, distinguishes among the 361 nonlinear codes $\mathcal C$ of kernel dimension $κ$ with $9\geqκ\geq 5$ obtained via Solov'eva-Phelps doubling construction. Each of the 361 resulting graphs has most of its nonloop edges expressible in terms of the lexicographically disjoint quarters of the products of the components of two of the ten 1-perfect partitions of length 8 classified by Phelps, and loops mostly expressible in terms of the lines of the Fano plane.

math.CO↗

arXiv2009

On a $\vec{C}_4$-ultrahomogeneous oriented graph

The notion of a $\mathcal C$-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and subsequently a strongly connected $\vec{C}_4$-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, 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 are replaced by ordered pencils.

math.CO↗

arXiv2009

SQS-graphs of Solov'eva-Phelps codes

A binary extended 1-perfect code $\mathcal C$ folds over its kernel via the Steiner quadruple systems associated with its codewords. The resulting folding, proposed as a graph invariant for $\mathcal C$, distinguishes among the 361 nonlinear codes $\mathcal C$ of kernel dimension $κ$ obtained via Solov'eva-Phelps doubling construction, where $9\geqκ\geq 5$. Each of the 361 resulting graphs has most of its nonloop edges expressible in terms of lexicographically ordered quarters of products of classes from extended 1-perfect partitions of length 8 (as classified by Phelps) and loops mostly expressible in terms of the lines of the Fano plane.

math.CO↗

arXiv2009

Quasiperfect domination in triangular lattices

A vertex subset $S$ of a graph $G$ is a perfect (resp. quasiperfect) dominating set in $G$ if each vertex $v$ of $G\setminus S$ is adjacent to only one vertex ($d_v\in\{1,2\}$ vertices) of $S$. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli symbol $\{3,6\}$ and in its toroidal quotients are investigated, yielding the classification of their perfect dominating sets and most of their quasiperfect dominating sets $S$ with induced components of the form $K_ν$, where $ν\in\{1,2,3\}$ depends only on $S$.

math.CO↗

arXiv2008

On a {K_4,K_{2,2,2}}-ultrahomogeneous graph

The existence of a connected 12-regular $\{K_4,K_{2,2,2}\}$-ultrahomogeneous graph $G$ is established, (i.e. each isomorphism between two copies of $K_4$ or $K_{2,2,2}$ in $G$ extends to an automorphism of $G$), with the 42 ordered lines of the Fano plane taken as vertices. This graph $G$ can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of $K_4$ and as the edge-disjoint union of 21 induced copies of $K_{2,2,2}$, with no more copies of $K_4$ or $K_{2,2,2}$ existing in $G$. Moreover, each edge of $G$ is shared by exactly one copy of $K_4$ and one of $K_{2,2,2}$. While the line graphs of $d$-cubes, ($3\le d\in\ZZ$), are $\{K_d, K_{2,2}\}$-ultrahomogeneous, $G$ is not even line-graphical. In addition, the chordless 6-cycles of $G$ are seen to play an interesting role and some self-dual configurations associated to $G$ with 2-arc-transitive, arc-transitive and semisymmetric Levi graphs are considered.

math.CO↗

arXiv2007

Perfect domination in rectangular grid graphs

A dominating set $S$ in a graph $G$ is said to be perfect if every vertex of $G$ not in $S$ is adjacent to just one vertex of $S$. Given a vertex subset $S'$ of a side $P_m$ of an $m\times n$ grid graph $G$, the perfect dominating sets $S$ in $G$ with $S'=S\cap V(P_m)$ can be determined via an exhaustive algorithm $Θ$ of running time $O(2^{m+n})$. Extending $Θ$ to infinite grid graphs of width $m-1$, periodicity makes the binary decision tree of $Θ$ 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 via $Θ$, for the growth and determination of which a speedier %greedy algorithm exists. %and their periodic structure, further studied. A recent characterization of grid graphs having total perfect codes $S$ (with just 1-cubes as induced components), due to Klostermeyer and Goldwasser, is given in terms of $Θ$, which allows to show that these sets $S$ are restrictions of only one total perfect code $S_1$ in the integer lattice graph $Λ$ of $\R^2$. Moreover, the complement $Λ-S_1$ yields an aperiodic tiling, like the Penrose tiling. In contrast, the parallel, horizontal, total perfect codes in $Λ$ are in 1-1 correspondence with the doubly infinite $\{0,1\}$-sequences.

math.CO↗