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Italo J. Dejter

Publications and source records attributed to Italo J. Dejter.

At least 19 recordsLinked to original sources
arXiv2026

Efficient total facial coloring of combinatorial maps

Let $1<k\in\mathbb{Z}$. A total coloring of a connected simple regular graph $G$ via color set $]k]_0=\{0,1,\ldots, k\}$ is efficient if each color yields an efficient dominating set, where efficient domination applies to each color class in $V(G)$. An efficient total coloring of $G$ via $[k]_0$ extends to an efficient total facial coloring $c$ of a map $M(G)$ of $G$ if it assigns a vertex-and-edge $k$-color set to the boundary cycle of each face, with the consequently missing color assigned to the face itself, so that the two adjacent faces along each edge are assigned different colors. Focus is set upon planar and toroidal maps. Modifications of $c$ are also treated by removing a perfect dominating set or a 1-factor from $G$.

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arXiv2026

List and total colorings of multiset permutation graphs

Let $k$ and $\ell$ be positive integers. The multiset star transposition graph ST$_k^\ell$ has as vertices the $k\ell$-strings $v_0\cdots v_{k\ell-1}$ on $k$ symbols, each symbol repeated $\ell$ times, and edges given by the transpositions $(v_0\;v_i)$ with $v_i\ne v_0$ ($0 1$ and $\ell>2$ that ST$_k^\ell$ is $(\ell-1)$-choosable and that, as a result, admits total colorings. In order to prove such assertions, the notion of efficient domination set (or E-set) of a graph is generalized for $\ell>1$ to that of an efficient dominating$\,^\ell$-set and applied to the graphs ST$_k^\ell$\,, showing they admit vertex partitions that generalize the Dejter-Serra partitions of ST$_k^1$ into E-sets, but not efficiently in the sense that the distance of each E$^\ell$-set be 3. Efficiently in such sense however, $ST^2_k$ and the related 2-set pancake permutation graph PC$^2_k$, among other intermediate permutation graphs, are shown to admit total colorings with $2k-1$ colors that determine partitions into $2k-1$ E-sets, each with distance 3. Furthermore, associated E-chains are examined.

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arXiv2026

Total coloring and efficient domination applications to non-Cayley non-Schreier vertex-transitive graphs

Let $0<k\in\mathbb{Z}$. Let the star 2-set transposition graph $ST^2_k$ be the $(2k-1)$-regular graph whose vertices are the $2k$-strings on $k$ symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such $2k$-string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph $PC^2_k$ has the same vertex set of $ST^2_k$ and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of $ST^2_k$. For $1<k\in\mathbb{Z}$, we show that $ST^2_k$ and $PC^2_k$, among other intermediate transposition graphs, have total colorings via $2k-1$ colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of $ST^2_k$ and $PC^2_k$, and partitions into into $2k-1$ such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.

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arXiv2025

Tight factorizations of girth-$g$-regular graphs

Girth-regular graphs with equal girth, regular degree and chromatic index are studied for the determination of 1-factorizations with each 1-factor intersecting every girth cycle. Applications to hamiltonian decomposability and to 3-dimensional geometry are given. Applications are suggested for priority assignment and optimization problems.

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arXiv2025

Universal updates of Dyck-nest signatures

Let $0<k\in\mathbb{Z}$. The anchored Dyck words of length $n=2k+1$ (obtained by prefixing a 0-bit to each Dyck word of length $2k$ and used to reinterpret the Hamilton cycles in the odd graph $O_k$ and the middle-levels graph $M_k$ found by Mütze et al.) represent in $O_k$ (resp., $M_k$) the cycles of an $n$- (resp., $2n$-) 2-factor and its cyclic (resp., dihedral) vertex classes, and are equivalent to Dyck-nest signatures. A sequence is obtained by updating these signatures according to the depth-first order of a tree of restricted growth strings (RGS's), reducing the RGS-generation of Dyck words by collapsing to a single update the time-consuming $i$-nested castling used to reach each non-root Dyck word or Dyck nest. This update is universal, for it does not depend on $k$.

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arXiv2025

Total coloring of regular graphs of girth = degree + 1

Let $2\le k\in\mathbb{Z}$. A total coloring of a $k$-regular simple graph via $k+1$ colors is an {\it efficient total coloring} if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. In this work, focus is set upon graphs of girth $k+1$. Efficient total colorings of finite connected simple cubic graphs of girth 4 are constructed starting at the 3-cube. It is conjectured that all of them are obtained by means of four basic operations. In contrast, the Robertson 19-vertex $(4,5)$-cage, the alternate union $Pet^k$ of a (Hamilton) $10k$-cycle with $k$ pentagon and $k$-pentagram $5$-cycles, for $k>1$ not divisible by 5, and its double cover $Dod^k$, contain TCs that are nonefficient. Applications to partitions into 3-paths and 3-stars are given.

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arXiv2025

Castling tree of tight Dyck nests with applications to odd and middle-levels graphs

A subfamily of Dyck words called tight Dyck words is seen to correspond, via a "castling" procedure, to the vertex set of an ordered tree $T$. From $T$, a "blowing" operation recreates the whole family ol Dyck words. The vertices of $T$ can be elementarily updated all along $T$. This simplifies an edge-supplementary arc-factorization view of Hamilton cycles of odd and middle-levels graphs found by T. Mütze et al. This take into account that the Dyck words represent: {\bf(a)} the cyclic and dihedral vertex classes of odd and middle-levels graphs, respectively, and {\bf(b)} the cycles of their 2-factors, as found by T. Mütze et al.

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arXiv2024

A numeral system for the middle-levels graphs

The middle-levels graph $M_k$ ($0<k\in\mathbb{Z}$) has a dihedral quotient pseudograph $R_k$ whose vertices are the $k$-edge ordered trees $T$, each $T$ encoded as a $(2k+1)$-string $F(T)$ formed via $\rightarrow$DFS by: {\bf(i)} ($\leftarrow$BFS-assigned) Kierstead-Trotter lexical colors $0,\ldots,k$ for the descending nodes; {\bf(ii)} asterisks $*$ for the $k$ ascending edges. Two ways of corresponding a restricted-growth $k$-string $α$ to each $T$ exist, namely one Stanley's way and a novel way that assigns $F(T)$ to $α$ via nested substring-swaps. These swaps permit to sort $V(R_k)$ as an ordered tree that allows a lexical visualization of $M_k$ as well as the Hamilton cycles of $M_k$ constructed by P. Gregor, T. Mütze and J. Nummenpalo.

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arXiv2024

Reinterpreting the Middle-Levels Theorem via Natural Enumeration of Ordered Trees

Let $0<k\in\mathbb{Z}$. A reinterpretation of the proof of existence of Hamilton cycles in the middle-levels graph $M_k$ induced by the vertices of the $(2k+1)$-cube representing the $k$- and $(k+1)$-subsets of $\{0,\ldots,2k\}$ is given via an associated dihedral quotient graph of $M_k$ whose vertices represent the ordered (rooted) trees of order $k+1$ and size $k$.

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arXiv2024

An alternate form of Merino-Mička-Mütze's approach to a combinatorial generation problem of Knuth

A modification of Merino-Mička-Mütze's solution to a combinatorial generation problem of Knuth is proposed in this survey. The resulting alternate form to such solution is compatible with a reinterpretation by the author of a proof of existence of Hamilton cycles in the middle-levels graphs. Such reinterpretation is given in terms of a dihedral quotient graph associated to each middle-levels graph. The vertices of such quotient graph represent Dyck words and their associated ordered trees. Those Dyck words are linearly ordered via a rooted tree that covers all their tight, or irreducible, forms, offering an universal reference point of view to express and integrate the periodic paths, or blocks, whose concatenation leads to Hamilton cycles resulting from the said solution.

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arXiv2024

A generalization of perfect codes in the presence of star multiset transpositions

Let $0<\ell\in\mathbb{Z}$. The notion of an efficient dominating set or perfect code $S$ of a graph $G$ is generalized to that of an efficient dominating$\,^\ell$-set or perfect$^\ell$code, of the graph $G$, meaning that each vertex $v$ of $V(G)\setminus S$ has exactly $\ell$ neighbors in $S$, instead of just one neighbor. Such generalization is applied to star $j$-set transposition graphs based on permutations of multisets with each symbol repeated $j$ times, ($j\in\{\ell,\ell-1\}$). In such vertex-transitive graphs this approach produces total colorings, efficient dominating sets, also called perfect codes, etc.

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arXiv2023

Arc coloring of odd graphs for hamiltonicity

Coloring the arcs of biregular graphs was introduced with possible applications to industrial chemistry, molecular biology, cellular neuroscience, etc. Here, we deal with arc coloring in some non-bipartite graphs. In fact, for $1<k\in\mathbb{Z}$, we find that the odd graph $O_k$ has an arc factorization with colors $0,1,\ldots,k$ such that the sum of colors of the two arcs of each edge equals $k$. This is applied to analyzing the influence of such arc factorizations in recently constructed uniform 2-factors in $O_k$ and in Hamilton cycles in $O_k$ as well as in its double covering graph known as the middle-levels graph $M_k$.

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arXiv2022

Representation of Dyck words in tensors that zipper merge contiguous integer compositions

Let $0<k\in\mathbb{Z}$. We zipper-merge integer compositions with sums $k$ and $k+1$, equal number of parts and initial entries equal at least to 1 and 2, respectively. This yields bitstrings with two initial zeros, $k-1$ remaining zeros and $k$ ones. Tensors whose entries are such bitstrings contain unique representations of all Dyck words of length $2k$. If rows and columns of such tensors are disposed in descending lexicographic order, then their entries not representing Dyck words form disjoint unions of descending staircases corresponding to strict lower triangular submatrices.

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arXiv2021

Homogeneous edge-disjoint $K_{2s}$ and $T_{st,t}$ unions

Let $r>2$ and $σ\in(0,r-1)$ be integers. We require $t<2s$, where $t=2^{σ+1}-1$ and $s=2^{r-σ-1}$. Generalizing a known $\{K_4,T_{6,3}\}$-ultrahomogenous graph $G_3^1$, we find that a finite, connected, undirected, arc-transitive graph $G_r^σ$ exists each of whose edges is shared by just two maximal subgraphs, namely a clique $X_0=K_{2s}$ and a $t$-partite regular-Turán graph $X_1=T_{st,t}$ on $s$ vertices per part. Each copy $Y$ of $X_i$ ($i=0,1$) in $G_r^σ$ shares each edge with just one copy of $X_{1-i}$ and all such copies of $X_{1-i}$ are pairwise distinct. Moreover, $G_r^σ$ is an edge-disjoint union of copies of $X_i$, for $i=0,1$. We prove that $G_r^σ$ is $\{K_{2s},T_{st,t}\}$-homogeneous if $t<2s$, and just $\{T_{st,t}\}$-homogeneous otherwise, meaning that there is an automorphism of $G_r^σ$ between any two such copies of $X_i$ relating two preselected arcs.

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arXiv2020

On Coloring the Arcs of Biregular Graphs

Recalling each edge of a graph $H$ has 2 oppositely oriented arcs, each vertex $v$ of $H$ is identified with the set of arcs, denoted $(v,e)$, departing from $v$ along the edges $e$ of $H$ incident to $v$. Let $H$ be a $(λ,μ)$-biregular graph with bipartition $(Y,X)$, where $|Y|=kμ$ and $|X|=kλ$, ($0<k,λ,μ\in\mathbb{Z}$). We consider the problem, for each edge $e=yx$ in $H$, of assigning, a color (given by an element) of $Y$, resp. $X$, to the arc $(y,e)$, resp. $(x,e)$, so that each color is assigned exactly once in the set of arcs departing from each vertex of $H$. Furthermore, we set such assignment to fulfill a specific bicolor weight function over a monotonic subset of $Y\times X$. This problem applies to the Design of Experiments for Industrial Chemistry, Molecular Biology, Cellular Neuroscience, etc. An algorithmic construction based on biregulzr graphs with bipartitions given by cyclic-group pairs is presented, as well as 3 essentially different solutions to the Great Circle Challenge Puzzle based on a different biregular graph whose bipartition is formed by the vertices and 5-cycles of the Petersen graph.

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