Strong geodetic number of complete bipartite graphs, crown graphs and hypercubes

The strong geodetic number, $\text{sg}(G),$ of a graph $G$ is the smallest number of vertices such that by fixing one geodesic between each pair of selected vertices, all vertices of the graph are covered. In this paper, the study of the strong geodetic number of complete bipartite graphs is continued. The formula for $\text{sg}(K_{n,m})$ is given, as well as a formula for the crown graphs $S_n^0$. Bounds on $\text{sg}(Q_n)$ are also discussed.Comment: 13 pages, 1 figure, 1 tabl

Graph theory general position problem

The classical no-three-in-line problem is to find the maximum number of points that can be placed in the $n \times n$ grid so that no three points lie on a line. Given a set $S$ of points in an Euclidean plane, the General Position Subset Selection Problem is to find a maximum subset $S'$ of $S$ such that no three points of $S'$ are collinear. Motivated by these problems, the following graph theory variation is introduced: Given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a gp-set of $G$ and its size is the gp-number ${\rm gp}(G)$ of $G$. Upper bounds on ${\rm gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete

Daisy Hamming graphs

Daisy graphs of a rooted graph $G$ with the root $r$ were recently introduced as a generalization of daisy cubes, a class of isometric subgraphs of hypercubes. In this paper we first solve the problem posed in \cite{Taranenko2020} and characterize rooted graphs $G$ with the root $r$ for which all daisy graphs of $G$ with respect to $r$ are isometric in $G$. We continue the investigation of daisy graphs $G$ (generated by $X$) of a Hamming graph $H$ and characterize those daisy graphs generated by $X$ of cardinality 2 that are isometric in $H$. Finally, we give a characterization of isometric daisy graphs of a Hamming graph $K_{k_1}\Box \ldots \Box K_{k_n}$ with respect to $0^n$ in terms of an expansion procedure

Weakly modular graphs and nonpositive curvature

This article investigates structural, geometrical, and topological characterizations and properties of weakly modular graphs and of cell complexes derived from them. The unifying themes of our investigation are various `nonpositive curvature' and `local-to-global' properties and characterizations of weakly modular graphs and their subclasses. Weakly modular graphs have been introduced as a far-reaching common generalization of median graphs (and more generally, of modular and orientable modular graphs), Helly graphs, bridged graphs, and dual polar graphs occurring under different disguises in several seemingly-unrelated fields of mathematics: Metric graph theory, Geometric group theory, Incidence geometries and buildings, Theoretical computer science and combinatorial optimization. We give a local-to-global characterization of weakly modular graphs and their subclasses in terms of simple connectedness of associated triangle-square complexes and specific local combinatorial conditions. In particular, we revisit characterizations of dual polar graphs by Cameron and by Brouwer-Cohen. We also show that (disk-)Helly graphs are precisely the clique-Helly graphs with simply connected clique complexes. With $l_1$-embeddable weakly modular and sweakly modular graphs we associate high-dimensional cell complexes, having several strong topological and geometrical properties (contractibility and the CAT(0) property). Their cells have a specific structure: they are basis polyhedra of even $\triangle$-matroids in the first case and orthoscheme complexes of gated dual polar subgraphs in the second case. We resolve some open problems concerning subclasses of weakly modular graphs: we prove a Brady-McCammond conjecture about CAT(0) metric on the orthoscheme complexes of modular lattices; we answer Chastand's question about prime graphs for pre-median graphs.Comment: to appear in Mem. Amer. Math. Soc.

On the isometric path partition problem

The isometric path cover (partition) problem of a graph is to find a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric $k$-path partition problem for $k\geq 3$ are NP-complete on general graphs. Fisher and Fitzpatrick \cite{FiFi01} have shown that the isometric path cover number of $(r\times r)$-dimensional grid is $\lceil 2r/3\rceil$. We show that the isometric path cover (partition) number of $(r\times s)$-dimensional grid is $s$ when $r \geq s(s-1)$. We establish that the isometric path cover (partition) number of $(r\times r)$-dimensional torus is $r$ when $r$ is even and is either $r$ or $r+1$ when $r$ is odd. Then, we demonstrate that the isometric path cover (partition) number of an $r$-dimensional Benes network is $2^r$. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids.Comment: This is a part of a proposed research project of Kuwait University, Kuwait. The author will appreciate to receive your comments and constructive criticism on this initial draft. The author's contact address is [email protected]

Strong geodetic number of complete bipartite graphs and of graphs with specified diameter

The strong geodetic problem is a recent variation of the classical geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on one fixed geodesic between a pair of vertices from $S$. In this paper, some general properties of the strong geodesic problem are studied, especially in connection with diameter of a graph. The problem is also solved for balanced complete bipartite graphs.Comment: 15 pages, 3 figures, 1 tabl

Golden codes: quantum LDPC codes built from regular tessellations of hyperbolic 4-manifolds

We adapt a construction of Guth and Lubotzky [arXiv:1310.5555] to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like $n^{0.1}$ where $n$ is the number of physical qubits. Similarly as in [arXiv:1310.5555], our homological code family stems from hyperbolic 4-manifolds equipped with tessellations. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.Comment: 30 pages, 4 figure

Nice labeling problem for event structures: a counterexample

In this note, we present a counterexample to a conjecture of Rozoy and Thiagarajan from 1991 (called also the nice labeling problem) asserting that any (coherent) event structure with finite degree admits a labeling with a finite number of labels, or equivalently, that there exists a function $f: \mathbb{N} \mapsto \mathbb{N}$ such that an event structure with degree $\le n$ admits a labeling with at most $f(n)$ labels. Our counterexample is based on the Burling's construction from 1965 of 3-dimensional box hypergraphs with clique number 2 and arbitrarily large chromatic numbers and the bijection between domains of event structures and median graphs established by Barth\'elemy and Constantin in 1993

Polychromatic Colorings on the Hypercube

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a structured class of colorings, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of a graph G contains every color into a question of coloring the 2-dimensional grid so that every so-called shape sequence corresponding to G contains every color. After surveying the tools for finding polychromatic numbers, we apply these techniques to find polychromatic numbers of a class of graphs called punctured hypercubes. We also consider the problem of finding polychromatic numbers in the setting where larger subcubes of the hypercube are colored. We exhibit two new constructions which show that this problem is not a straightforward generalization of the edge coloring problem.Comment: 24 page

A Counterexample to Thiagarajan\u27s Conjecture on Regular Event Structures

We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular prime event structures correspond exactly to those obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded natural-cliques are recognizable by finite trace automata. Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these structures as combinatorial and geometric objects and both conjectures can be reformulated in this framework. Namely, the domains of prime event structures correspond exactly to pointed median graphs; from a geometric point of view, these domains are in bijection with pointed CAT(0) cube complexes. A necessary condition for both conjectures to be true is that domains of respective regular event structures admit a regular nice labeling. To disprove these conjectures, we describe a regular event domain (with bounded natural-cliques) that does not admit a regular nice labeling. Our counterexample is derived from an example by Wise (1996 and 2007) of a nonpositively curved square complex whose universal cover is a CAT(0) square complex containing a particular plane with an aperiodic tiling