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Bobby Miraftab

Publications and source records attributed to Bobby Miraftab.

15 recordsLinked to original sources
arXiv2026

Hyperfiniteness of boundary actions via tree decompositions

We study conditions for a countable group acting on a connected locally finite hyperbolic graph to induce a hyperfinite orbit equivalence relation on the Gromov boundary of the graph in terms of tree-decompositions of the graph. We prove that for a connected locally finite hyperbolic graph $X$ equipped with an action of a countable group $G$, if $(T, \beta)$ is a $G$-invariant tree-decomposition of $X$ such that each bag induces a connected subgraph $X_t$ of $X$ for each $t \in V(T)$, each adhesion set is finite and such that there are only finitely many $G$-orbits of edges of $T$, then the orbit equivalence relation of $G$ acting on the Gromov boundary $\partial X$ is hyperfinite provided the orbit equivalence relation of $G$ acting on $\partial T$ is hyperfinite and the orbit equivalence relations of the bag stabilizers acting on $\partial X_t$ are all hyperfinite. We show that the converse also holds if $(T, \beta)$ satisfies the additional property that each adhesion set distinguishes at least two ends of $X$.

math.GR↗
arXiv2026

On the Spanning Ratio of the Greedy Triangulation for Convex Point Sets

The greedy triangulation of a finite planar point set is obtained by considering all segments in nondecreasing order of length and inserting each segment that does not cross an earlier one. Its spanning ratio is known to be bounded by a universal constant, but the standard bound obtained from the diamond and good-polygon properties is about $11739.1$. We prove a substantially smaller bound for points in convex position. In particular, for every finite point set $P\subset\mathbb{R}^2$ in convex position and every pair $u,v\in P$, the greedy triangulation contains a $u$--$v$ path of length at most $\kappa |uv|$, where $\kappa<17.814$. Thus, the greedy triangulation of a convex point set is an $18$-spanner.

cs.CG↗
arXiv2026

Quasirandomness and Uniform Twin-Width

For every nontrivial finite group, we prove that its quasirandom degree gives a polynomial lower bound on its uniform twin-width, whereas its minimum faithful complex representation degree gives a linear upper bound. For nonabelian finite simple groups, these two parameters coincide, so uniform twin-width is polynomially equivalent to quasirandomness in that class, yielding a new definition of quasirandomness in the sense of Gowers. We use the lower bound to prove that uniform twin-width is unbounded over finite groups, which helps us construct finitely presented groups with finite twin-width but infinite uniform twin-width. This answers a question of Bonnet, Geniet, Tessera and Thomasse. Finally, we determine the uniform twin-width of all three Thompson groups.

math.GR↗
arXiv2026

The Volume Helly Theorem in the plane, colorful version

We prove a colorful volume Helly theorem for convex sets in $\mathbb R^2$: There is a constant $V>0$ such that if $\mathfrak F_1,\mathfrak F_2,\mathfrak F_3,\mathfrak F_4$ are finite families of convex sets in $\mathbb R^2$ and if $|\bigcap_1^4F_i|\ge V$ for every transversal $F_i\in \mathfrak F_i,\; (i=1,\ldots,4)$, then $|\bigcap \mathfrak{F}_i|\ge 1$ for some $i$. Here $|A|$ is the Lebesgue measure of $A\subset \mathbb R^d$. The main ingredient is the following theorem. Let $Q_1,\ldots,Q_4\subset\mathbb R^2$ be convex quadrilaterals of area at most $1$, where of course each $Q_i$ is the intersection of 4 halfplanes. Then for every $Q_i$ there is one of these halfplanes $H_i$, say, such that $|\bigcap_1^4 H_i| \le 4096$.

math.CO↗
arXiv2026

Automorphism Groups in Extremal Families of Polyhedral Graphs

We study automorphism groups in five extremal families of polyhedral graphs. For every $n\ge14$, we prove that every minimum-order $3$-polytopal graph containing a vertex of each degree $3,4,\ldots,n$ is asymmetric. The proof uses an exact planar defect decomposition, a complete description of the high-degree tail, and a saturation theorem for the subgraph induced by the uniquely high-degree vertices. Duality gives the corresponding asymmetry result for minimum-face polyhedra containing faces of every size $3,4,\ldots,n$. For the three polyhedral graphs whose complements are also polyhedral, we determine the ordinary and extended automorphism groups and identify the extended group \[ \mathsf{Aut}^{\pm}(G_{13})\cong (C_2\times C_2)\rtimes C_4. \] Next, we classify automorphism groups of radius-one polyhedra. In the unique-dominating-vertex case they are cyclic or dihedral, and in the triangulated case the possibilities are \[ 1,\qquad C_2,\qquad C_3,\qquad C_2\times C_2,\qquad S_3. \] For polyhedra that are unigraphic among the class of self-dual, we show that their automorphism group is either $1$ or $C_2$. Finally, we consider polyhedra that are products of graphs, for each of the four standard graph products, and we classify them according to their automorphism group.

math.CO↗
arXiv2026

On $\varepsilon$-Matrix Product Factorization of graphs

We introduce an approximate version of matrix product factorization for graphs. A simple graph $G$ on $n$ vertices is said to admit an $\varepsilon$-matrix product factorization if there exist simple graphs $H$ and $K$ on the same vertex set such that $A(H)A(K)$ and $A(G)$ disagree in at most $\varepsilon n^{2}$ entries. This Hamming-type relaxation preserves, outside the error set, the exact interpretation of each edge as having a unique $H$-then-$K$ two-step witness. We establish equivalent matrix, and witness formulations, showing that the sets $N_H(w)\times N_K(w)$ form an approximate disjoint decomposition of the ordered adjacency relation of $G$, and we derive quantitative constraints involving walk counts and the degrees of the factor graphs. We then construct approximate factorizations for several graph families. Every complete graph $K_n$ has matrix-product-factorization distance $O(1/n)$, despite the exact congruence obstruction that permits exact factorization only when $n\equiv 1\pmod 4$. More generally, a blow-up of a fixed graph on $r$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le r/n$, and the construction is exact whenever every non-isolated part has even order. For bipartite graphs, we give one-sided factorizations that realize one orientation of almost all edges. In particular, every tree on $n\ge2$ vertices admits an $\varepsilon$-factorization with $\varepsilon\le 1/n$, although no nontrivial tree is exactly factorizable. These results show that rigid exact obstructions may disappear under a vanishing proportion of entrywise errors.

math.CO↗
arXiv2026

Cayley Graphs Of Order $pqrs$ Are Hamiltonian

Assume $ G $ is a finite group with order $ |G| = pqrs $, where $ p $, $ q $, $ r $, and $ s $ are distinct prime numbers. We prove that every connected Cayley graph of $ G $ contains a hamiltonian cycle. Our result drops all restrictions of all previously known results on hamiltonian cycles in Cayley graphs of groups of order $pqrs$.

math.CO↗
arXiv2026

On Matrix Product Factorization in Association Schemes

We study matrix product factorizations (MPFs) in symmetric association schemes: identities $A_SA_T=A_U$ where $A_S,A_T,A_U$ are loopless unions of basic relations and the ordinary matrix product is again a $0$-$1$ adjacency matrix. We give equivalent structural and spectral criteria for MPFs, derive valency and rank restrictions, and analyze several standard families. For $2$-class schemes, the only nontrivial loopless MPF comes from the scheme of the $5$-cycle. For $P$-polynomial schemes, the distance-regular recurrence gives strong restrictions on products $A_1A_i$. We also prove a universal pentagon theorem for the case $A_SA_T=J-I$, and show that extremal rank forces all non-zero eigenvalues of $A_U$ to be $\pm k(U)$, hence gives bipartiteness. Finally, in Hamming schemes we obtain rank obstructions and classify MPFs of the form $A_1A_T=A_U$: in $H(d,2)$, for $d\ge2$, the only non-zero loopless example is $A_1A_d=A_{d-1}$, which is trivial since $A_d$ has valency $1$; for $q>2$, no non-zero example occurs.

math.CO↗
arXiv2026

Accessibility and Twin-width

We show that finite twin-width does not imply accessibility for finitely generated groups, which answers a question of Esperet. That is, we prove that there exists a finitely generated group $\Gamma$ that has finite uniform twin-width but is not accessible. In particular, for every finite generating set $S$ of $\Gamma$, the Cayley graph $Cay(\Gamma ; S)$ has finite twin-width but is not accessible. The example is obtained by combining Wilkes construction of a finitely generated inaccessible residually $p$-finite groups with a result of Bonnet, Geniet, Tessera and Thomasse regarding the twin-width of groups acting faithfully on regular rooted trees.

math.GR↗
arXiv2026

3-packings in Triangulations: Algorithms, bounds, and Complexity

We study $H$-packings in plane triangulations for the three-vertex graphs $H\in\{P_3,K_3,P_2\cup P_1\}$. For a graph $H$, let $\lambda_H(G)$ denote the maximum size of an $H$-packing in $G$, with the convention that for $H=P_2\cup P_1$ the copies are required to be induced. For $P_3$-packings, we prove that every triangulation $G$ on $n$ vertices satisfies $\lambda_{P_3}(G)\ge \left\lfloor \frac n5\right\rfloor$, and show that this lower bound is asymptotically tight. We also study triangle packings in triangulations and provide lower bounds for $\lambda_{K_3}(G)$ in terms of the maximum degree and the degree sequence. We give a face-path characterization of triangle factors in $4$-connected plane triangulations using a hamiltonian cycle and the weak duals of the two associated maximal outerplanar graphs. Finally, for induced packings by $P_2\cup P_1$, we prove that every plane triangulation $T$ on $n$ vertices satisfies $\lambda_{P_2\cup P_1}(T)\ge \left\lfloor \frac n3\right\rfloor-2$, and show that such a packing can be found in polynomial time.

cs.DM↗
arXiv2026

Vertex cuts and median decompositions

Median decompositions were introduced by Stavropoulos in 2015 as a generalisation of tree decompositions. In this paper, we further develop and exposit this theory as a tool in structural graph theory to study systems of vertex separations. Generalising the well-known fact that nested systems of vertex separations produce tree decompositions of a graph over the structure tree, we describe how a (not necessarily nested) system of separations produces a median decomposition. The median graph in this decomposition is the `dual median graph' constructed by Sageev. If the system of cuts is nested then this median decomposition recovers precisely the aforementioned tree decomposition. We prove a theorem asserting that this decomposition is `uniquely minimal', and describe how Sageev--Roller duality manifests in median decompositions. As an application of our structural approach, we extend a theorem of Stavropoulos from finite graphs to all graphs, which states that the median-width a graph is equal to its clique number. We also describe the link between (canonical) median decompositions and (equivariant) coarse embeddings/quasi-isometries into median graphs. A corollary of these results is a characterisation of when a finitely generated group acts metrically-properly/geometrically on a median graph, in terms of canonical median decompositions of its Cayley graphs.

math.CO↗
arXiv2026

2-dimensional unit vector flows

We study $2$-dimensional unit vector flows on graphs, that is, nowhere-zero flows that assign to each oriented edge a unit vector in $\mathbb R^{3}$. We give a new geometric characterization of $\mathbb S^{2}$-flows on cubic graphs. We also prove that the class of cubic graphs admitting an $\mathbb S^{2}$-flow is closed under a natural composition operation, which yields further constructions; in particular, blowing up a vertex into a triangle preserves the existence of an $\mathbb S^{2}$-flow. Our second contribution is algebraic: we extend the rank-based approach of [SIAM J. Discrete Math., 29 (2015), pp.~2166--2178] from $\mathbb S^{1}$-flows to $\mathbb S^{2}$-flows. More precisely, we show that if an $\mathbb S^{2}$-flow $\varphi$ satisfies $\operatorname{rank}(S_{\mathbb{Q}}(\varphi))\le 2$ and $S_{\mathbb{Q}}(\varphi)$ is odd-coordinate-free, then the graph admits a nowhere-zero $4$-flow.

math.CO↗
arXiv2025

On Prime Matrix Product Factorizations

A graph $G$ factors into graphs $H$ and $K$ via a matrix product if $A = BC$, where $A$, $B$, and $C$ are the adjacency matrices of $G$, $H$, and $K$, respectively. The graph $G$ is prime if, in every such factorization, one of the factors is a perfect matching that is, it corresponds to a permutation matrix. We characterize all prime graphs, then using this result we classify all factorable forests, answering a question of Akbari et al. [\emph{Linear Algebra and its Applications} (2025)]. We prove that every torus is factorable, and we characterize all possible factorizations of grids, addressing two questions posed by Maghsoudi et al. [\emph{Journal of Algebraic Combinatorics} (2025)].

math.CO↗
arXiv2025

On Matrix Product Factorization of Cayley graphs

We study when the adjacency matrix of a Cayley graph factors as the product of two adjacency matrices of Cayley graphs. Let $G$ be a finite group and let $U\subseteq G\setminus \{e\}$ be symmetric. Writing $A(G;U)$ for the adjacency matrix of the Cayley graph of $G$ with respect to $U$, we prove that for symmetric subsets $S,T,U$ of $G\setminus \{e\}$, $A(G;U)=A(G;S)\,A(G;T)$ if and only if $U=ST$ and each $u\in U$ has a unique representation $u=st$, equivalently $\bigl(\sum_{s\in S}s\bigr)\bigl(\sum_{t\in T}t\bigr)=\sum_{u\in U}u$ in the group algebra. When $S,T,U$ are unions of conjugacy classes, this is characterized character-theoretically by $\chi(U)=\chi(S)\chi(T)/\chi(1)$ for all $\chi\in\mathrm{Irr}(G)$. In addition, for abelian groups, we identify $A(G;S)A(G;T)$ with the $0\!-\!1$ convolution $\mathbf{1}_S*\mathbf{1}_T$, so factorability is equivalent to $(S,T)$ being a Sidon pair, i.e., $(S-S)\cap(T-T)=\{0\}$. For cyclic groups, we reformulate factorability via mask polynomials and reduce to prime-power components using the Chinese Remainder Theorem. We also analyze dihedral groups $D_{2n}$, presenting infinite families of factorable generating sets, and give explicit constructions of subsets whose Cayley graphs do and do not admit such factorizations.

math.CO↗