Geometry of Partial Cubes
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Hypercellular Graphs: Partial Cubes Without Q As Partial Cube Minor
− Hypercellular graphs: partial cubes without Q3 as partial cube minor Victor Chepoi1, Kolja Knauer1;2, and Tilen Marc3 1Laboratoire d'Informatique et Syst`emes,Aix-Marseille Universit´eand CNRS, Facult´edes Sciences de Luminy, F-13288 Marseille Cedex 9, France fvictor.chepoi, [email protected] 2 Departament de Matem`atiquesi Inform`atica,Universitat de Barcelona (UB), Barcelona, Spain 3 Faculty of Mathematics and Physics, University of Ljubljana and Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia [email protected] Abstract. We investigate the structure of isometric subgraphs of hypercubes (i.e., partial cubes) which do not contain − finite convex subgraphs contractible to the 3-cube minus one vertex Q3 (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). Extending similar results for median and cellular graphs, we show that the convex hull of an isometric cycle of such a graph is gated and isomorphic to the Cartesian product of edges and even cycles. Furthermore, we show that our graphs are exactly the class of partial cubes in which any finite convex subgraph can be obtained from the Cartesian products of edges and even cycles via successive gated amalgams. This decomposition result enables us to establish a variety of results. In particular, it yields that our class of graphs generalizes median and cellular graphs, which motivates naming our graphs hypercellular. Furthermore, we show that hypercellular graphs are tope graphs of zonotopal complexes of oriented matroids. Finally, we characterize hypercellular graphs as being median-cell { a property naturally generalizing the notion of median graphs. -
Daisy Cubes and Distance Cube Polynomial
Daisy cubes and distance cube polynomial Sandi Klavˇzar a,b,c Michel Mollard d July 21, 2017 a Faculty of Mathematics and Physics, University of Ljubljana, Slovenia [email protected] b Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia c Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia d Institut Fourier, CNRS Universit´eGrenoble Alpes, France [email protected] Dedicated to the memory of our friend Michel Deza Abstract n Let X ⊆ {0, 1} . The daisy cube Qn(X) is introduced as the subgraph of n Qn induced by the union of the intervals I(x, 0 ) over all x ∈ X. Daisy cubes are partial cubes that include Fibonacci cubes, Lucas cubes, and bipartite wheels. If u is a vertex of a graph G, then the distance cube polynomial DG,u(x,y) is introduced as the bivariate polynomial that counts the number of induced subgraphs isomorphic to Qk at a given distance from the vertex u. It is proved that if G is a daisy cube, then DG,0n (x,y)= CG(x + y − 1), where CG(x) is the previously investigated cube polynomial of G. It is also proved that if G is a daisy cube, then DG,u(x, −x) = 1 holds for every vertex u in G. Keywords: daisy cube; partial cube; cube polynomial; distance cube polynomial; Fibonacci cube; Lucas cube AMS Subj. Class. (2010): 05C31, 05C75, 05A15 1 Introduction In this paper we introduce a class of graphs, the members of which will be called daisy cubes. -
Edge General Position Problem Is to find a Gpe-Set
Edge General Position Problem Paul Manuela R. Prabhab Sandi Klavˇzarc,d,e a Department of Information Science, College of Computing Science and Engineering, Kuwait University, Kuwait [email protected] b Department of Mathematics, Ethiraj College for Women, Chennai, Tamilnadu, India [email protected] c Faculty of Mathematics and Physics, University of Ljubljana, Slovenia [email protected] d Faculty of Natural Sciences and Mathematics, University of Maribor, Slovenia e Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia Abstract Given a graph G, the general position problem is to find a largest set S of vertices of G such that no three vertices of S lie on a common geodesic. Such a set is called a gp-set of G and its cardinality is the gp-number, gp(G), of G. In this paper, the edge general position problem is introduced as the edge analogue of the general position problem. The edge general position number, gpe(G), is the size of a largest edge r general position set of G. It is proved that gpe(Qr)=2 and that if T is a tree, then gpe(T ) is the number of its leaves. The value of gpe(Pr Ps) is determined for every r, s ≥ 2. To derive these results, the theory of partial cubes is used. Mulder’s meta-conjecture on median graphs is also discussed along the way. Keywords: general position problem; edge general position problem; hypercube; tree; grid graph; partial cube; median graph AMS Subj. Class. (2020): 05C12, 05C76 arXiv:2105.04292v1 [math.CO] 10 May 2021 1 Introduction The geometric concept of points in general position, the still open Dudeney’s no-three- in-line problem [6] from 1917 (see also [18, 22]), and the general position subset selection problem [8, 28] from discrete geometry, all motivated the introduction of a related concept in graph theory as follows [19]. -
Isometric Diamond Subgraphs
Isometric Diamond Subgraphs David Eppstein Computer Science Department, University of California, Irvine [email protected] Abstract. We test in polynomial time whether a graph embeds in a distance- preserving way into the hexagonal tiling, the three-dimensional diamond struc- ture, or analogous higher-dimensional structures. 1 Introduction Subgraphs of square or hexagonal tilings of the plane form nearly ideal graph draw- ings: their angular resolution is bounded, vertices have uniform spacing, all edges have unit length, and the area is at most quadratic in the number of vertices. For induced sub- graphs of these tilings, one can additionally determine the graph from its vertex set: two vertices are adjacent whenever they are mutual nearest neighbors. Unfortunately, these drawings are hard to find: it is NP-complete to test whether a graph is a subgraph of a square tiling [2], a planar nearest-neighbor graph, or a planar unit distance graph [5], and Eades and Whitesides’ logic engine technique can also be used to show the NP- completeness of determining whether a given graph is a subgraph of the hexagonal tiling or an induced subgraph of the square or hexagonal tilings. With stronger constraints on subgraphs of tilings, however, they are easier to con- struct: one can test efficiently whether a graph embeds isometrically onto the square tiling, or onto an integer grid of fixed or variable dimension [7]. In an isometric em- bedding, the unweighted distance between any two vertices in the graph equals the L1 distance of their placements in the grid. An isometric embedding must be an induced subgraph, but not all induced subgraphs are isometric. -
Cubic Vertex-Transitive Graphs of Girth Six
CUBIC VERTEX-TRANSITIVE GRAPHS OF GIRTH SIX PRIMOZˇ POTOCNIKˇ AND JANOSˇ VIDALI Abstract. In this paper, a complete classification of finite simple cubic vertex-transitive graphs of girth 6 is obtained. It is proved that every such graph, with the exception of the Desargues graph on 20 vertices, is either a skeleton of a hexagonal tiling of the torus, the skeleton of the truncation of an arc-transitive triangulation of a closed hyperbolic surface, or the truncation of a 6-regular graph with respect to an arc- transitive dihedral scheme. Cubic vertex-transitive graphs of girth larger than 6 are also discussed. 1. Introduction Cubic vertex-transitive graph are one of the oldest themes in algebraic graph theory, appearing already in the classical work of Foster [13, 14] and Tutte [33], and retaining the attention of the community until present times (see, for example, the works of Coxeter, Frucht and Powers [8], Djokovi´cand Miller [9], Lorimer [23], Conder and Lorimer [6], Glover and Maruˇsiˇc[15], Potoˇcnik, Spiga and Verret [27], Hua and Feng [16], Spiga [30], to name a few of the most influential papers). The girth (the length of a shortest cycle) is an important invariant of a graph which appears in many well-known graph theoretical problems, results and formulas. In many cases, requiring the graph to have small girth severely restricts the structure of the graph. Such a phenomenon can be observed when one focuses to a family of graphs of small valence possessing a high level of symmetry. For example, arc-transitive 4-valent graphs of girth at most 4 were characterised in [29]. -
The Distance-Regular Antipodal Covers of Classical Distance-Regular Graphs
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 52. COMBINATORICS, EGER (HUNGARY), 198'7 The Distance-regular Antipodal Covers of Classical Distance-regular Graphs J. T. M. VAN BON and A. E. BROUWER 1. Introduction. If r is a graph and "Y is a vertex of r, then let us write r,("Y) for the set of all vertices of r at distance i from"/, and r('Y) = r 1 ('Y)for the set of all neighbours of 'Y in r. We shall also write ry ...., S to denote that -y and S are adjacent, and 'YJ. for the set h} u r b) of "/ and its all neighbours. r i will denote the graph with the same vertices as r, where two vertices are adjacent when they have distance i in r. The graph r is called distance-regular with diameter d and intersection array {bo, ... , bd-li c1, ... , cd} if for any two vertices ry, oat distance i we have lfH1(1')n nr(o)j = b, and jri-ib) n r(o)j = c,(o ::; i ::; d). Clearly bd =co = 0 (and C1 = 1). Also, a distance-regular graph r is regular of degree k = bo, and if we put ai = k- bi - c; then jf;(i) n r(o)I = a, whenever d("f, S) =i. We shall also use the notations k, = lf1b)I (this is independent of the vertex ry),). = a1,µ = c2 . For basic properties of distance-regular graphs, see Biggs [4]. The graph r is called imprimitive when for some I~ {O, 1, ... , d}, I-:/= {O}, If -:/= {O, 1, .. -
Math.RA] 25 Sep 2013 Previous Paper [3], Also Relying in Conceptually Separated Tools from Them, Such As Graphs and Digraphs
Certain particular families of graphicable algebras Juan Núñez, María Luisa Rodríguez-Arévalo and María Trinidad Villar Dpto. Geometría y Topología. Facultad de Matemáticas. Universidad de Sevilla. Apdo. 1160. 41080-Sevilla, Spain. [email protected] [email protected] [email protected] Abstract In this paper, we introduce some particular families of graphicable algebras obtained by following a relatively new line of research, ini- tiated previously by some of the authors. It consists of the use of certain objects of Discrete Mathematics, mainly graphs and digraphs, to facilitate the study of graphicable algebras, which are a subset of evolution algebras. 2010 Mathematics Subject Classification: 17D99; 05C20; 05C50. Keywords: Graphicable algebras; evolution algebras; graphs. Introduction The main goal of this paper is to advance in the research of a novel mathematical topic emerged not long ago, the evolution algebras in general, and the graphicable algebras (a subset of them) in particular, in order to obtain new results starting from those by Tian (see [4, 5]) and others already obtained by some of us in a arXiv:1309.6469v1 [math.RA] 25 Sep 2013 previous paper [3], also relying in conceptually separated tools from them, such as graphs and digraphs. Concretely, our goal is to find some particular types of graphicable algebras associated with well-known types of graphs. The motivation to deal with evolution algebras in general and graphicable al- gebras in particular is due to the fact that at present, the study of these algebras is very booming, due to the numerous connections between them and many other branches of Mathematics, such as Graph Theory, Group Theory, Markov pro- cesses, dynamic systems and the Theory of Knots, among others. -
Arxiv:2011.14609V1 [Math.CO] 30 Nov 2020 Vertices in Different Partition Sets Are Linked by a Hamilton Path of This Graph
Symmetries of the Honeycomb toroidal graphs Primož Šparla;b;c aUniversity of Ljubljana, Faculty of Education, Ljubljana, Slovenia bUniversity of Primorska, Institute Andrej Marušič, Koper, Slovenia cInstitute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia Abstract Honeycomb toroidal graphs are a family of cubic graphs determined by a set of three parameters, that have been studied over the last three decades both by mathematicians and computer scientists. They can all be embedded on a torus and coincide with the cubic Cayley graphs of generalized dihedral groups with respect to a set of three reflections. In a recent survey paper B. Alspach gathered most known results on this intriguing family of graphs and suggested a number of research problems regarding them. In this paper we solve two of these problems by determining the full automorphism group of each honeycomb toroidal graph. Keywords: automorphism; honeycomb toroidal graph; cubic; Cayley 1 Introduction In this short paper we focus on a certain family of cubic graphs with many interesting properties. They are called honeycomb toroidal graphs, mainly because they can be embedded on the torus in such a way that the corresponding faces are hexagons. The usual definition of these graphs is purely combinatorial where, somewhat vaguely, the honeycomb toroidal graph HTG(m; n; `) is defined as the graph of order mn having m disjoint “vertical” n-cycles (with n even) such that two consecutive n-cycles are linked together by n=2 “horizontal” edges, linking every other vertex of the first cycle to every other vertex of the second one, and where the last “vertcial” cycle is linked back to the first one according to the parameter ` (see Section 3 for a precise definition). -
Graphs and Groups Ainize Cidoncha Markiegui
Graphs and Groups Final Degree Dissertation Degree in Mathematics Ainize Cidoncha Markiegui Supervisor: Gustavo Adolfo Fern´andezAlcober Leioa, 1 September 2014 Contents Introduction v 1 Automorphisms of graphs 1 1.1 Graphs . .1 1.2 Automorphisms . .3 1.3 Actions of groups on sets and the Orbit- Stabilizer Theorem . .7 1.3.1 Actions on graphs . 10 1.4 The automorphism group of a cyclic graph . 11 2 Automorphism groups of Kneser graphs 13 2.1 Kneser graphs . 13 2.2 The automorphism group of all Kneser graphs . 14 2.2.1 The Erd}os-Ko-Radotheorem . 14 2.2.2 The automorphism group of Kneser graphs . 19 3 Automorphism groups of generalized Petersen graphs 21 3.1 The generalized Petersen graphs . 21 3.2 Automorphism groups of generalized Petersen graphs . 24 3.2.1 The subgroup B(n; k).................. 25 3.2.2 The automorphism group A(n; k)............ 30 4 An application of graphs and groups to reaction graphs 39 4.1 Molecular graphs and rearrangements . 39 4.2 Reaction graphs . 42 4.2.1 1,2-shift in the carbonium ion . 43 A Solved exercises 49 A.1 Chapter 1 . 49 A.2 Chapter 2 . 51 A.3 Chapter 3 . 57 A.4 Chapter 4 . 66 Bibliography 71 iii Introduction Graph theory has a wide variety of research fields, such as in discrete math- ematics, optimization or computer sciences. However, this work will be focused on the algebraic branch of graph theory. Number and group theory are necessary in order to develop this project whose aim is to give enough de- tails and clarifications in order to fully understand the meaning and concept of the automorphism group of a graph. -
Arxiv:2010.05518V3 [Math.CO] 28 Jan 2021 Consists of the Fibonacci Strings of Length N, N Fn = {V1v2
FIBONACCI-RUN GRAPHS I: BASIC PROPERTIES OMER¨ EGECIO˘ GLU˘ AND VESNA IRSIˇ Cˇ Abstract. Among the classical models for interconnection networks are hyper- cubes and Fibonacci cubes. Fibonacci cubes are induced subgraphs of hypercubes obtained by restricting the vertex set to those binary strings which do not con- tain consecutive 1s, counted by Fibonacci numbers. Another set of binary strings which are counted by Fibonacci numbers are those with a restriction on the run- lengths. Induced subgraphs of the hypercube on the latter strings as vertices define Fibonacci-run graphs. They have the same number of vertices as Fibonacci cubes, but fewer edges and different connectivity properties. We obtain properties of Fibonacci-run graphs including the number of edges, the analogue of the fundamental recursion, the average degree of a vertex, Hamil- tonicity, and special degree sequences, the number of hypercubes they contain. A detailed study of the degree sequences of Fibonacci-run graphs is interesting in its own right and is reported in a companion paper. Keywords: Hypercube, Fibonacci cube, Fibonacci number. AMS Math. Subj. Class. (2020): 05C75, 05C30, 05C12, 05C40, 05A15 1. Introduction The n-dimensional hypercube Qn is the graph on the vertex set n f0; 1g = fv1v2 : : : vn j vi 2 f0; 1gg; where two vertices v1v2 : : : vn and u1u2 : : : un are adjacent if vi 6= ui for exactly one index i 2 [n]. In other words, vertices of Qn are all possible strings of length n consisting only of 0s and 1s, and two vertices are adjacent if and only if they differ n n−1 in exactly one coordinate or \bit". -
5 Partial Cubes
5 Partial Cubes This is the central chapter of the book. Unlike general cubical graphs, iso- metric subgraphs of cubes—partial cubes—can be effectively characterized in many different ways and possess much finer structural properties. In this chapter, we present what can be called a “micro” theory of partial cubes. 5.1 Definitions and Examples As cubical graphs are subgraphs of cubes, partial cubes are isometric sub- graphs of cubes (cf. Section 1.4). Definition 5.1. A graph G is a partial cube if it can be isometrically embed- ded into a cube H(X) for some set X. We often identify G with its isometric image in H(X) and say that G is a partial cube on the set X. Note that partial cubes are connected graphs. Clearly, they are cubical graphs. The converse is not true. The smallest counterexample is the graph G on seven vertices shown in Figure 5.1a. z {a,b,c} x y {a,b} {b,c} a) b) a c u v w { } {b} { } G s Ø Figure 5.1. A cubical graph that is not a partial cube. S. Ovchinnikov, Graphs and Cubes, Universitext, DOI 10.1007/978-1-4614-0797-3_5, 127 © Springer Science+Business Media, LLC 2011 128 5 Partial Cubes Indeed, suppose that ϕ : G → H(X) is an embedding. We may assume that ϕ(s)= ∅. Then the images of vertices u, v, and w under the embedding ϕ are singletons in X, say, {a}, {b}, and {c}, respectively. It is easy to verify that we must have (cf. -
Eigenvalues and Distance-Regularity of Graphs Edwin Van
EvDRG EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation Eigenvalues and distance-regularity of graphs Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Edwin van Dam Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Dept. Econometrics and Operations Research Remove vertices Remove edges Tilburg University Adding edges Amalgamate Generalized Odd Proof Graph Theory and Interactions, Durham, July 20, 2013 Durham, July 20, 2013 Edwin van Dam – 1 / 24 Dedication EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation Structure Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues Partial linear space q-ary Desargues Ugly DRGs Perturbations Remove vertices Remove edges Adding edges Amalgamate Generalized Odd Proof David Gregory Durham, July 20, 2013 Edwin van Dam – 2 / 24 Spectrum EvDRG Dedication Spectrum Two many Distance-regular Walks Central equation A (finite simple) graph Γ on n vertices Structure Twisted and odd Good conditions Polynomials Projection ⇑ ? Spectral Excess ⇓ Desargues Partial linear space q-ary Desargues Ugly DRGs The spectrum (of eigenvalues) λ ≥ ... ≥ λ Perturbations 1 n Remove vertices of the (a) 01-adjacency matrix A of Γ Remove edges Adding edges Amalgamate Generalized Odd Proof Durham, July 20, 2013 Edwin van Dam – 3 / 24 Two many EvDRG Dedication Spectrum Two many Distance-regular There are 2 graphs on 30 vertices with spectrum Walks Central equation Structure 12, 2 (9×), 0 (15×), −6 (5×). Twisted and odd Good conditions Polynomials Projection Spectral Excess Desargues There are more than 60,000 graphs on 30 vertices with spectrum Partial linear space q-ary Desargues 12, 3 (10×), 0 (5×), −3 (14×).