Fibonacci-Run Graphs I: Basic Properties
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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". -
Structure of Fibonacci Cubes: a Survey
Structure of Fibonacci cubes: a survey Sandi Klav·zar Faculty of Mathematics and Physics University of Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics University of Maribor, Slovenia e-mail: [email protected] Abstract The Fibonacci cube ¡n is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as intercon- nection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper a survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results. Their median nature that leads to a fast recognition algorithm is discussed. The Fibonacci dimension of a graph, studies of graph invariants on Fibonacci cubes, and related classes of graphs are also presented. Along the way some new short proofs are given. Key words: Fibonacci cube; Fibonacci number; Cartesian product of graphs; median graph; degree sequence; cube polynomial AMS Subj. Class.: 05-02, 05C75, 05C45, 05C07, 05C12. 1 Introduction Let B = f0; 1g and for n ¸ 1 set Bn = fb1b2 : : : bn j bi 2 B; 1 · i · ng : The n-cube Qn is the graph de¯ned on the vertex set Bn, vertices b1b2 : : : bn and 0 0 0 0 b1b2 : : : bn being adjacent if bi 6= bi holds for exactly one i 2 f1; : : : ; ng. Hypercubes, as the n-cubes are also called, form one of the central classes in graph theory. On one hand they are important from the theoretical point of view, on the other hand they form a model for numerous applications. -
The Second Malta Conference in Graph Theory and Combinatorics 2017
The Second Malta Conference in Graph Theory and Combinatorics 2017 2MCGTC-2017 in honour of the 75th birthday of Professor Stanley Fiorini 26{30 June 2017 ii Welcome Address Mer~ba! We are honoured that you chose to join us for The Second Malta Confer- ence in Graph Theory and Combinatorics. This conference is commemorating the 75th birthday of Professor Stanley Fiorini, who introduced graph theory and combinatorics at the University of Malta. Many of you may be asking when the previous Malta Conference was held? The First Malta Conference on Graphs and Combinatorics was held during the period 28 May { 2 June, 1990, at the Suncrest Hotel, also in Qawra, St Paul's Bay. It differed from a number of similar conferences held in the central Mediterranean region at that time in that it consisted of three types of lectures. L´aszl´oLov´aszand Carsten Thomassen delivered two instructional courses of five one-hour lectures each; the former was A survey of independent sets in graphs and the latter was on Embeddings of graphs. There were also four invited speakers, namely L.W. Beineke, N.L. Biggs, R. Graham and D.J.A. Welsh, each of whom gave a one-hour lecture. The third type of talks were the 20-minute contributed talks running in two parallel sessions and given by 39 speakers. Volume 124 (1994) of the journal Discrete Mathematics was a special edition dedicated to this conference; it was edited by Stanley Fiorini and Josef Lauri, and it consisted of 22 selected papers. Twenty-seven years later we are gathered here for the second such conference organ- ised in the Island of Malta. -
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". -
The Fibonacci Dimension of a Graph
The Fibonacci dimension of a graph Sergio Cabello∗ David Eppsteiny Sandi Klavˇzarz November 7, 2018 Abstract The Fibonacci dimension fdim(G) of a graph G is introduced as the smallest integer f such that G admits an isometric embedding into Γf , the f-dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a graph in terms of the isometric and lattice dimension, provide a combinatorial characterization of the Fibonacci dimension using properties of an associated graph, and establish the Fibonacci dimension for certain families of graphs. From the algorithmic point of view we prove that it is NP-complete to decide if fdim(G) equals to the isometric dimension of G, and that it is also NP-hard to approximate fdim(G) within (741=740) − ". We also give a (3=2)-approximation algorithm for fdim(G) in the general case and a (1 + ")-approximation algorithm for simplex graphs. 1 Introduction Hypercubes play a prominent role in metric graph theory as well as in several other areas such as parallel computing and coding theory. One of their central features is the ability to compute distances very efficiently because the distance between two vertices is simply the number of coordinates in which they differ; the same ability to compute distances may be transferred to any isometric subgraph of a hypercube. In this way partial cubes appear, a class of graphs intensively studied so far; see the books [12, 18, 31], the recent papers [3, 24, 42, 43], the recent (semi-)survey [41], and references therein. In particular we point out a recent fast recognition algorithm [17] and improvements in classification of cubic partial cubes [16, 36]. -
Double Graphsଁ Emanuele Munarini, Claudio Perelli Cippo, Andrea Scagliola, Norma Zagaglia Salvi
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 308 (2008) 242–254 www.elsevier.com/locate/disc Double graphsଁ Emanuele Munarini, Claudio Perelli Cippo, Andrea Scagliola, Norma Zagaglia Salvi Politecnico di Milano, Dipartimento di Matematica, Piazza Leonardo da Vinci 32, 20133 Milano, Italy Received 1 December 2004; received in revised form 11 November 2005; accepted 27 November 2006 Available online 25 May 2007 Abstract In this paper we study the elementary properties of double graphs, i.e. of graphs which are the direct product of a simple graph G with the graph obtained by the complete graph K2 adding a loop to each vertex. © 2007 Elsevier B.V. All rights reserved. MSC: 05C15; 05C50; 05C75 Keywords: Direct product; Lexicographic product; Cospectral graphs; Strongly regular graphs; Complexity; Graph homomorphisms; Retract; Chromatic index; Fibonacci cubes 1. Introduction In [18] it was observed that the binary strings of length n + 1 without zigzags, i.e. without 010 and 101 as factors, can be reduced to the Fibonacci strings, i.e. binary strings without two consecutive 1’s, of length n. The set of Fibonacci strings can be endowed with a graph structure saying that two strings are adjacent when they differ exactly in one position. The graphs obtained in this way are called Fibonacci cubes [12] and have been studied in several recent papers. We wondered if the set of all binary strings without zigzags could be endowed with some graph structure related in some way with Fibonacci cubes. -
Graph Theory, an Antiprism Graph Is a Graph That Has One of the Antiprisms As Its Skeleton
Graph families From Wikipedia, the free encyclopedia Chapter 1 Antiprism graph In the mathematical field of graph theory, an antiprism graph is a graph that has one of the antiprisms as its skeleton. An n-sided antiprism has 2n vertices and 4n edges. They are regular, polyhedral (and therefore by necessity also 3- vertex-connected, vertex-transitive, and planar graphs), and also Hamiltonian graphs.[1] 1.1 Examples The first graph in the sequence, the octahedral graph, has 6 vertices and 12 edges. Later graphs in the sequence may be named after the type of antiprism they correspond to: • Octahedral graph – 6 vertices, 12 edges • square antiprismatic graph – 8 vertices, 16 edges • Pentagonal antiprismatic graph – 10 vertices, 20 edges • Hexagonal antiprismatic graph – 12 vertices, 24 edges • Heptagonal antiprismatic graph – 14 vertices, 28 edges • Octagonal antiprismatic graph– 16 vertices, 32 edges • ... Although geometrically the star polygons also form the faces of a different sequence of (self-intersecting) antiprisms, the star antiprisms, they do not form a different sequence of graphs. 1.2 Related graphs An antiprism graph is a special case of a circulant graph, Ci₂n(2,1). Other infinite sequences of polyhedral graph formed in a similar way from polyhedra with regular-polygon bases include the prism graphs (graphs of prisms) and wheel graphs (graphs of pyramids). Other vertex-transitive polyhedral graphs include the Archimedean graphs. 1.3 References [1] Read, R. C. and Wilson, R. J. An Atlas of Graphs, Oxford, England: Oxford University Press, 2004 reprint, Chapter 6 special graphs pp. 261, 270. 2 1.4. EXTERNAL LINKS 3 1.4 External links • Weisstein, Eric W., “Antiprism graph”, MathWorld. -
The Fibonacci Dimension of a Graph
The Fibonacci dimension of a graph Sergio Cabello∗ David Eppstein† Sandi Klavˇzar‡ Submitted: Mar 13, 2009; Accepted: Feb 28, 2011; Published: Mar 11, 2011 Mathematics Subject Classification: 05C12, 05C75, 05C85 Abstract The Fibonacci dimension fdim(G) of a graph G is introduced as the smallest integer f such that G admits an isometric embedding into Γf , the f-dimensional Fibonacci cube. We give bounds on the Fibonacci dimension of a graph in terms of the isometric and lattice dimension, provide a combinatorial characterization of the Fibonacci dimension using properties of an associated graph, and establish the Fibonacci dimension for certain families of graphs. From the algorithmic point of view, we prove that it is NP-complete to decide whether fdim(G) equals the isometric dimension of G, and show that no algorithm to approximate fdim(G) has approx- imation ratio below 741/740, unless P=NP. We also give a (3/2)-approximation algorithm for fdim(G) in the general case and a (1+ ε)-approximation algorithm for simplex graphs. 1 Introduction Hypercubes play a prominent role in metric graph theory as well as in several other areas such as parallel computing and coding theory. One of their central features is the ability to compute distances very efficiently because the distance between two vertices is simply the number of coordinates in which they differ; the same ability to compute distances may be transferred to any isometric subgraph of a hypercube. In this way partial cubes appear, a class of graphs intensively studied so far; see the books [12, 19, 32], the recent papers [3, 25, 43, 44], the recent (semi-)survey [42], and references therein.