Constructions for Cubic Graphs with Large Girth
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Maximizing the Order of a Regular Graph of Given Valency and Second Eigenvalue∗
SIAM J. DISCRETE MATH. c 2016 Society for Industrial and Applied Mathematics Vol. 30, No. 3, pp. 1509–1525 MAXIMIZING THE ORDER OF A REGULAR GRAPH OF GIVEN VALENCY AND SECOND EIGENVALUE∗ SEBASTIAN M. CIOABA˘ †,JACKH.KOOLEN‡, HIROSHI NOZAKI§, AND JASON R. VERMETTE¶ Abstract. From Alon√ and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ<2 k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs. Key words. second eigenvalue, regular graph, expander AMS subject classifications. 05C50, 05E99, 68R10, 90C05, 90C35 DOI. 10.1137/15M1030935 1. Introduction. For a k-regular graph G on n vertices, we denote by λ1(G)= k>λ2(G) ≥ ··· ≥ λn(G)=λmin(G) the eigenvalues of the adjacency matrix of G. For a general reference on the eigenvalues of graphs, see [8, 17]. The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1, 8, 23], for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana and of Serre√ [1, 11, 15, 23, 24, 27, 30, 34, 35, 40], we know that v(k, λ) is finite for λ<2 k − 1. The recent result of Marcus, Spielman, and Srivastava [28] showing the existence of infinite families of√ Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 k − 1. -
The Intriguing Human Preference for a Ternary Patterned Reality
75 Kragujevac J. Sci. 27 (2005) 75-114. THE INTRIGUING HUMAN PREFERENCE FOR A TERNARY PATTERNED REALITY Lionello Pogliani,* Douglas J. Klein,‡ and Alexandru T. Balaban¥ *Dipartimento di Chimica Università della Calabria, 87030 Rende (CS), Italy; ‡Texas A&M University at Galveston, Galveston, Texas, 77553-1675, USA; ¥ Texas A&M University at Galveston, Galveston, Texas, 77551, USA E-mails: [email protected]; [email protected]; balabana @tamug.tamu.edu (Received February 10, 2005) Three paths through the high spring grass / One is the quicker / and I take it (Haikku by Yosa Buson, Spring wind on the river bank Kema) How did number theory degrade to numerology? How did numerology influence different aspects of human creation? How could some numbers assume such an extraordinary meaning? Why some numbers appear so often throughout the fabric of reality? Has reality really a preference for a ternary pattern? Present excursion through the ‘life’ and ‘deeds’ of this pattern and of its parent number in different fields of human culture try to offer an answer to some of these questions giving at the same time a glance of the strange preference of the human mind for a patterned reality based on number three, which, like the 'unary' and 'dualistic' patterned reality, but in a more emphatic way, ended up assuming an archetypical character in the intellectual sphere of humanity. Introduction Three is not only the sole number that is the sum of all the natural numbers less than itself, but it lies between arguably the two most important irrational and transcendental numbers of all mathematics, i.e., e < 3 < π. -
On the Cycle Double Cover Problem
On The Cycle Double Cover Problem Ali Ghassâb1 Dedicated to Prof. E.S. Mahmoodian Abstract In this paper, for each graph , a free edge set is defined. To study the existence of cycle double cover, the naïve cycle double cover of have been defined and studied. In the main theorem, the paper, based on the Kuratowski minor properties, presents a condition to guarantee the existence of a naïve cycle double cover for couple . As a result, the cycle double cover conjecture has been concluded. Moreover, Goddyn’s conjecture - asserting if is a cycle in bridgeless graph , there is a cycle double cover of containing - will have been proved. 1 Ph.D. student at Sharif University of Technology e-mail: [email protected] Faculty of Math, Sharif University of Technology, Tehran, Iran 1 Cycle Double Cover: History, Trends, Advantages A cycle double cover of a graph is a collection of its cycles covering each edge of the graph exactly twice. G. Szekeres in 1973 and, independently, P. Seymour in 1979 conjectured: Conjecture (cycle double cover). Every bridgeless graph has a cycle double cover. Yielded next data are just a glimpse review of the history, trend, and advantages of the research. There are three extremely helpful references: F. Jaeger’s survey article as the oldest one, and M. Chan’s survey article as the newest one. Moreover, C.Q. Zhang’s book as a complete reference illustrating the relative problems and rather new researches on the conjecture. A number of attacks, to prove the conjecture, have been happened. Some of them have built new approaches and trends to study. -
The Travelling Salesman Problem in Bounded Degree Graphs*
The Travelling Salesman Problem in Bounded Degree Graphs? Andreas Bj¨orklund1, Thore Husfeldt1, Petteri Kaski2, and Mikko Koivisto2 1 Lund University, Department of Computer Science, P.O.Box 118, SE-22100 Lund, Sweden, e-mail: [email protected], [email protected] 2 Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, P.O.Box 68, FI-00014 University of Helsinki, Finland, e-mail: [email protected], Abstract. We show that the travelling salesman problem in bounded- degree graphs can be solved in time O((2 − )n), where > 0 depends only on the degree bound but not on the number of cities, n. The algo- rithm is a variant of the classical dynamic programming solution due to Bellman, and, independently, Held and Karp. In the case of bounded in- teger weights on the edges, we also present a polynomial-space algorithm with running time O((2 − )n) on bounded-degree graphs. 1 Introduction There is no faster algorithm known for the travelling salesman problem than the classical dynamic programming solution from the early 1960s, discovered by Bellman [2, 3], and, independently, Held and Karp [9]. It runs in time within a polynomial factor of 2n, where n is the number of cities. Despite the half a cen- tury of algorithmic development that has followed, it remains an open problem whether the travelling salesman problem can be solved in time O(1.999n) [15]. In this paper we provide such an upper bound for graphs with bounded maximum vertex degree. For this restricted graph class, previous attemps have succeeded to prove such bounds when the degree bound, ∆, is three or four. -
Large Girth Graphs with Bounded Diameter-By-Girth Ratio 3
LARGE GIRTH GRAPHS WITH BOUNDED DIAMETER-BY-GIRTH RATIO GOULNARA ARZHANTSEVA AND ARINDAM BISWAS Abstract. We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girth Γk k → ∞ as k and diam Γ 6 D for some D > 0 and all k N. For each dimension n > 2, we find a → ∞ girth Γk ∈ pair of matrices in SLn(Z) such that (i) they generate a free subgroup, (ii) their reductions mod p generate SLn(Fp) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp) have girth at least cn log p for some cn > 0. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most O(log p). This gives infinite sequences of finite 4-regular Cayley graphs as in the title. These are the first explicit examples in all dimensions n > 2 (all prior examples were in n = 2). Moreover, they happen to be expanders. Together with Margulis’ and Lubotzky-Phillips-Sarnak’s classical con- structions, these new graphs are the only known explicit large girth Cayley graph expanders with bounded diameter-by-girth ratio. 1. Introduction The girth of a graph is the edge-length of its shortest non-trivial cycle (it is assigned to be infinity for an acyclic graph). The diameter of a graph is the greatest edge-length distance between any pair of its vertices. We regard a graph Γ as a sequence of its connected components Γ = (Γk)k∈N each of which is endowed with the edge-length distance. -
On Snarks That Are Far from Being 3-Edge Colorable
On snarks that are far from being 3-edge colorable Jonas H¨agglund Department of Mathematics and Mathematical Statistics Ume˚aUniversity SE-901 87 Ume˚a,Sweden [email protected] Submitted: Jun 4, 2013; Accepted: Mar 26, 2016; Published: Apr 15, 2016 Mathematics Subject Classifications: 05C38, 05C70 Abstract In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover. 1 Introduction A cubic graph is said to be colorable if it has a 3-edge coloring and uncolorable otherwise. A snark is an uncolorable cubic cyclically 4-edge connected graph. It it well known that an edge minimal counterexample (if such exists) to some classical conjectures in graph theory, such as the cycle double cover conjecture [15, 18], Tutte's 5-flow conjecture [19] and Fulkerson's conjecture [4], must reside in this family of graphs. There are various ways of measuring how far a snark is from being colorable. One such measure which was introduced by Huck and Kochol [8] is the oddness. The oddness of a bridgeless cubic graph G is defined as the minimum number of odd components in any 2-factor in G and is denoted by o(G). -
Girth in Graphs
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF COMBINATORIAL THEORY. Series B 35, 129-141 (1983) Girth in Graphs CARSTEN THOMASSEN Mathematical institute, The Technicai University oJ Denmark, Building 303, Lyngby DK-2800, Denmark Communicated by the Editors Received March 31, 1983 It is shown that a graph of large girth and minimum degree at least 3 share many properties with a graph of large minimum degree. For example, it has a contraction containing a large complete graph, it contains a subgraph of large cyclic vertex- connectivity (a property which guarantees, e.g., that many prescribed independent edges are in a common cycle), it contains cycles of all even lengths module a prescribed natural number, and it contains many disjoint cycles of the same length. The analogous results for graphs of large minimum degree are due to Mader (Math. Ann. 194 (1971), 295-312; Abh. Math. Sem. Univ. Hamburg 31 (1972), 86-97), Woodall (J. Combin. Theory Ser. B 22 (1977), 274-278), Bollobis (Bull. London Math. Sot. 9 (1977), 97-98) and Hlggkvist (Equicardinal disjoint cycles in sparse graphs, to appear). Also, a graph of large girth and minimum degree at least 3 has a cycle with many chords. An analogous result for graphs of chromatic number at least 4 has been announced by Voss (J. Combin. Theory Ser. B 32 (1982), 264-285). 1, INTRODUCTION Several authors have establishedthe existence of various configurations in graphs of sufficiently large connectivity (independent of the order of the graph) or, more generally, large minimum degree (see, e.g., [2, 91). -
Complexity and Approximation Results for the Connected Vertex Cover Problem in Graphs and Hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot
Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot To cite this version: Bruno Escoffier, Laurent Gourvès, Jérôme Monnot. Complexity and approximation results forthe connected vertex cover problem in graphs and hypergraphs. 2007. hal-00178912 HAL Id: hal-00178912 https://hal.archives-ouvertes.fr/hal-00178912 Preprint submitted on 12 Oct 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Laboratoire d'Analyse et Modélisation de Systèmes pour l'Aide à la Décision CNRS UMR 7024 CAHIER DU LAMSADE 262 Juillet 2007 Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier, Laurent Gourvès, Jérôme Monnot Complexity and approximation results for the connected vertex cover problem in graphs and hypergraphs Bruno Escoffier∗ Laurent Gourv`es∗ J´erˆome Monnot∗ Abstract We study a variation of the vertex cover problem where it is required that the graph induced by the vertex cover is connected. We prove that this problem is polynomial in chordal graphs, has a PTAS in planar graphs, is APX-hard in bipartite graphs and is 5/3-approximable in any class of graphs where the vertex cover problem is polynomial (in particular in bipartite graphs). -
Computing the Girth of a Planar Graph in Linear Time∗
Computing the Girth of a Planar Graph in Linear Time∗ Hsien-Chih Chang† Hsueh-I Lu‡ April 22, 2013 Abstract The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n5/4 log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time. 1 Introduction Let G be an edge-weighted simple graph, i.e., G does not contain multiple edges and self- loops. We say that G is unweighted if the weight of each edge of G is one. A cycle of G is simple if each node and each edge of G is traversed at most once in the cycle. The girth of G, denoted girth(G), is the minimum weight of all simple cycles of G. For instance, the girth of each graph in Figure 1 is four. As shown by, e.g., Bollobas´ [4], Cook [12], Chandran and Subramanian [10], Diestel [14], Erdos˝ [21], and Lovasz´ [39], girth is a fundamental combinatorial characteristic of graphs related to many other graph properties, including degree, diameter, connectivity, treewidth, and maximum genus. We address the problem of computing the girth of an n-node graph. Itai and Rodeh [28] gave the best known algorithm for the problem, running in time O(M(n) log n), where M(n) is the time for multiplying two n × n matrices [13]. -
Switching 3-Edge-Colorings of Cubic Graphs Arxiv:2105.01363V1 [Math.CO] 4 May 2021
Switching 3-Edge-Colorings of Cubic Graphs Jan Goedgebeur Department of Computer Science KU Leuven campus Kulak 8500 Kortrijk, Belgium and Department of Applied Mathematics, Computer Science and Statistics Ghent University 9000 Ghent, Belgium [email protected] Patric R. J. Osterg˚ard¨ Department of Communications and Networking Aalto University School of Electrical Engineering P.O. Box 15400, 00076 Aalto, Finland [email protected] In loving memory of Johan Robaey Abstract The chromatic index of a cubic graph is either 3 or 4. Edge- Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic graphs up to order 30 and bipartite cubic graphs up to order 36 are tabulated. Families of cubic graphs of orders 4n + 2 and 4n + 4 with 2n edge-Kempe equivalence classes are presented; it is conjectured that there are no cubic graphs arXiv:2105.01363v1 [math.CO] 4 May 2021 with more edge-Kempe equivalence classes. New families of nonplanar bipartite cubic graphs with exactly one edge-Kempe equivalence class are also obtained. Edge-Kempe switching is further connected to cycle switching of Steiner triple systems, for which an improvement of the established classification algorithm is presented. Keywords: chromatic index, cubic graph, edge-coloring, edge-Kempe switch- ing, one-factorization, Steiner triple system. 1 1 Introduction We consider simple finite undirected graphs without loops. For such a graph G = (V; E), the number of vertices jV j is the order of G and the number of edges jEj is the size of G. -
Bounded Treewidth
Algorithms for Graphs of (Locally) Bounded Treewidth by MohammadTaghi Hajiaghayi A thesis presented to the University of Waterloo in ful¯llment of the thesis requirement for the degree of Master of Mathematics in Computer Science Waterloo, Ontario, September 2001 c MohammadTaghi Hajiaghayi, 2001 ° I hereby declare that I am the sole author of this thesis. I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the purpose of scholarly research. MohammadTaghi Hajiaghayi I further authorize the University of Waterloo to reproduce this thesis by photocopying or other means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly research. MohammadTaghi Hajiaghayi ii The University of Waterloo requires the signatures of all persons using or photocopying this thesis. Please sign below, and give address and date. iii Abstract Many real-life problems can be modeled by graph-theoretic problems. These graph problems are usually NP-hard and hence there is no e±cient algorithm for solving them, unless P= NP. One way to overcome this hardness is to solve the problems when restricted to special graphs. Trees are one kind of graph for which several NP-complete problems can be solved in polynomial time. Graphs of bounded treewidth, which generalize trees, show good algorithmic properties similar to those of trees. Using ideas developed for tree algorithms, Arnborg and Proskurowski introduced a general dynamic programming approach which solves many problems such as dominating set, vertex cover and independent set. Others used this approach to solve other NP-hard problems. -
The Line Mycielskian Graph of a Graph
[VOLUME 6 I ISSUE 1 I JAN. – MARCH 2019] e ISSN 2348 –1269, Print ISSN 2349-5138 http://ijrar.com/ Cosmos Impact Factor 4.236 THE LINE MYCIELSKIAN GRAPH OF A GRAPH Keerthi G. Mirajkar1 & Veena N. Mathad2 1Department of Mathematics, Karnatak University's Karnatak Arts College, Dharwad - 580001, Karnataka, India 2Department of Mathematics, University of Mysore, Manasagangotri, Mysore-06, India Received: January 30, 2019 Accepted: March 08, 2019 ABSTRACT: In this paper, we introduce the concept of the line mycielskian graph of a graph. We obtain some properties of this graph. Further we characterize those graphs whose line mycielskian graph and mycielskian graph are isomorphic. Also, we establish characterization for line mycielskian graphs to be eulerian and hamiltonian. Mathematical Subject Classification: 05C45, 05C76. Key Words: Line graph, Mycielskian graph. I. Introduction By a graph G = (V,E) we mean a finite, undirected graph without loops or multiple lines. For graph theoretic terminology, we refer to Harary [3]. For a graph G, let V(G), E(G) and L(G) denote the point set, line set and line graph of G, respectively. The line degree of a line uv of a graph G is the sum of the degree of u and v. The open-neighborhoodN(u) of a point u in V(G) is the set of points adjacent to u. For each ui of G, a new point uˈi is introduced and the resulting set of points is denoted by V1(G). Mycielskian graph휇(G) of a graph G is defined as the graph having point set V(G) ∪V1(G) ∪v and line set E(G) ∪ {xyˈ : xy∈E(G)} ∪ {y ˈv : y ˈ ∈V1(G)}.