arXiv subjects

Thomas Zaslavsky

Publications and source records attributed to Thomas Zaslavsky.

At least 19 recordsLinked to original sources

arXiv2026

On strongly regular signed graphs of higher girth

Strongly regular signed graphs are an extension of strongly regular graphs to the realm of signed graphs, that is, graphs where each edge is positive or negative. Unlike with ordinary strongly regular graphs, most kinds of signed counterparts with girth 4 or higher are describable in terms of known structures. We prove that those with girth 4 that are bipartite are classified by designs of two kinds: weighing matrix designs and symmetric block designs. Those of girth 5 are few and readily described. There are none of higher girth. Those with girth 4 that are not bipartite are unsolved.

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arXiv2026

Obstructions to Total Rainbow Forests in Edge-Colored Graphs

A total rainbow forest in an edge-colored graph is a forest that contains every edge color exactly once. Using a necessary and sufficient condition that a total rainbow forest exists, we demonstrate the existence of huge numbers of edge-colored graphs that are minimal obstructions to such existence.

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arXiv2025

The Cycle Counts of Graphs

We prove that an inseparable graph can have any positive number of cycles with the six exceptions 2, 4, 5, 8, 9, 16, and that an inseparable cubic graph has the additional exceptions 1 and 13. The exceptions for simple inseparable cubic graphs are unknown.

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arXiv2025

Loose elements in binary and ternary matroids

We call a matroid element "loose" if it is contained in no circuits of size less than the rank of the matroid. A matroid in which all elements are loose is a paving matroid. Acketa determined all binary paving matroids, while Oxley specified all ternary paving matroids. We characterize the binary matroids that contain a loose element. For ternary matroids with a loose element, we show that their size is linear in terms of their rank. Moreover, for a prime power $q$, we give a partial characterization of $GF(q)$-representable matroids that have two or more loose elements; we note Rajpal's partial characterization of $GF(q)$-representable paving matroids as a consequence.

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arXiv2024

Apex Graphs and Cographs

A class $\mathcal{G}$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $\mathcal{G}^\mathrm{apex}$ the class of graphs $G$ that contain a vertex $v$ such that $G-v$ is in $\mathcal{G}$. We prove that if a hereditary class $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does $\mathcal{G}^\mathrm{apex}$. The hereditary class of cographs consists of all graphs $G$ that can be generated from $K_1$ using complementation and disjoint union. A graph is an apex cograph if it contains a vertex whose deletion results in a cograph. Cographs are precisely the graphs that do not have the $4$-vertex path as an induced subgraph. Our main result finds all such forbidden induced subgraphs for the class of apex cographs.

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arXiv2024

Projective Rectangles: A New Kind of Incidence Structure

A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial Desargues's theorem, a construction from projective planes, and alternative formulations. In sequels we study harmonic conjugation and the graphs of lines and subplanes.

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arXiv2024

Projective Rectangles: Harmonic Conjugation

A projective rectangle is like a projective plane that has different lengths in two directions. We develop harmonic conjugation in projective rectangles. We construct projective rectangles in some harmonic matroids (matroids where harmonic conjugation is defined on every collinear point triple), such as Desarguesian projective planes of finite characteristic, by harmonic conjugation from extended lift matroids based on finite fields. Similar results follow for countable fields with characteristic $0$. We also show that projective rectangles are almost harmonic matroids.

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arXiv2024

Projective Rectangles: The Graph of Lines

A projective rectangle is like a projective plane that may have different lengths in two directions. We develop properties of the graph of lines, in which adjacency means having a common point, especially its strong regularity and clique structure. The main construction of projective rectangles, stated in a previous paper, gives rectangles whose graph of lines is a known strongly regular bilinear forms graph. That fact leads to a proof that the main construction does produce projective rectangles, and also gives a new representation of bilinear forms graphs. We conclude by mentioning a few simple graph properties, such as the chromatic number, which is not known, and a partial geometry obtained from the graph.

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arXiv2024

Weakly Negative Circles Versus Best Clustering in Signed Graphs

Clustering a signed graph means partitioning the vertices into sets ("clusters") so that every positive edge, and no negative edge, is within a cluster. Clustering is not always possible; the obstruction is circles with exactly one negative edge ("weakly negative circles"). The correlation clustering problem is to cluster with the minimum number of edges that violate the clustering rule, called $Q$. A lower bound is $w$, the maximum number of edge-disjoint weakly negative circles. If every two such circles are edge disjoint, then $Q=w$. We characterize signed graphs of this kind.

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arXiv2024

Cobiased graphs: Single-element extensions and elementary quotients of graphic matroids

Zaslavsky (1991) introduced a graphical structure called a biased graph and used it to characterize all single-element coextensions and elementary lifts of graphic matroids. We introduce a new, dual graphical structure that we call a cobiased graph and use it to characterize single-element extensions and elementary quotients of graphic matroids.

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arXiv2024

The Rhodes semilattice of a biased graph

We reinterpret the Rhodes semilattices $R_n(\mathfrak{G})$ of a group $\mathfrak{G}$ in terms of gain graphs and generalize them to all gain graphs, both as sets of partition-potential pairs and as sets of subgraphs, and for the latter, further to biased graphs. Based on this we propose four different natural lattices in which the Rhodes semilattices and its generalizations are order ideals.

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arXiv2023

Whitney Numbers of Partial Dowling Lattices

The Dowling lattice $Q_n(\mathfrak{G})$, $\mathfrak{G}$ a finite group, generalizes the geometric lattice generated by all vectors, over a field, with at most two nonzero components. Abstractly, it is a fundamental object in the classification of finite matroids. Constructively, it is the frame matroid of a certain gain graph known as $\mathfrak{G}{\cdot}K_n^{(V)}$. Its Whitney numbers of the first kind enter into several important formulas. Ravagnani suggested and partially proved that these numbers of $Q_n(\mathfrak{G})$ and higher-weight generalizations are polynomial functions of $|\mathfrak{G}|$. We give a simple proof for $Q_n(\mathfrak{G})$ and its generalization to a wider class of gain graphs and biased graphs, and we determine the degrees and coefficients of the polynomials.

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arXiv2023

Vector Valued Switching in Signed Graphs

A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of switching to higher dimensions. Using this concept, we define balancing dimension and strong balancing dimension for a signed graph, which can be used for a new classification of degree of imbalance of unbalanced signed graphs. We provide bounds for the balancing and strong balancing dimensions, and calculate these dimensions for some classes of signed graphs.

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arXiv2022

Matroids of Gain Signed Graphs

A signed graph has edge signs. A gain graph has oriented edge gains drawn from a group. We define the combination of the two for the abelian case, in which each oriented edge of a signed graph has a gain from an abelian group, concentrating on the case of the additive group of a field. We develop the elementary graph properties, the associated matroid, and the vector and hyperplanar representations.

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arXiv2021

Characterizations of Some Parity Signed Graphs

We describe parity labelings of signed graphs; equivalently, cuts of the underlying graph that have nearly equal sides. We characterize the balanced signed graphs which are parity signed graphs. We give structural characterizations of all parity signed stars, bistars, cycles, paths and complete bipartite graphs. The rna number of a graph is the smallest cut size that has nearly equal sides; we find it for a few classes of parity signed graphs.

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arXiv2021

Total graph of a signed graph

The total graph is built by joining the graph to its line graph by means of the incidences. We introduce a similar construction for signed graphs. Under two similar definitions of the line signed graph, we define the corresponding total signed graph and we show that it is stable under switching. We consider balance, the frustration index and frustration number, and the largest eigenvalue. In the regular case we compute the spectrum of the adjacency matrix of the total graph and the spectra of certain compositions, and we determine some with exactly two main eigenvalues.

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arXiv2021

Two Hamiltonian cycles

If the line graph of a graph $G$ decomposes into Hamiltonian cycles, what is $G$? We answer this question for decomposition into two cycles.

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arXiv2020

Signed Distance Laplacian Matrices for Signed Graphs

A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs.

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