COMBINATORICS of EMBEDDINGS 2 Triangulated by a Simplicial Complex K Is Denoted |K|
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Investigations on Unit Distance Property of Clebsch Graph and Its Complement
Proceedings of the World Congress on Engineering 2012 Vol I WCE 2012, July 4 - 6, 2012, London, U.K. Investigations on Unit Distance Property of Clebsch Graph and Its Complement Pratima Panigrahi and Uma kant Sahoo ∗yz Abstract|An n-dimensional unit distance graph is on parameters (16,5,0,2) and (16,10,6,6) respectively. a simple graph which can be drawn on n-dimensional These graphs are known to be unique in the respective n Euclidean space R so that its vertices are represented parameters [2]. In this paper we give unit distance n by distinct points in R and edges are represented by representation of Clebsch graph in the 3-dimensional Eu- closed line segments of unit length. In this paper we clidean space R3. Also we show that the complement of show that the Clebsch graph is 3-dimensional unit Clebsch graph is not a 3-dimensional unit distance graph. distance graph, but its complement is not. Keywords: unit distance graph, strongly regular The Petersen graph is the strongly regular graph on graphs, Clebsch graph, Petersen graph. parameters (10,3,0,1). This graph is also unique in its parameter set. It is known that Petersen graph is 2-dimensional unit distance graph (see [1],[3],[10]). It is In this article we consider only simple graphs, i.e. also known that Petersen graph is a subgraph of Clebsch undirected, loop free and with no multiple edges. The graph, see [[5], section 10.6]. study of dimension of graphs was initiated by Erdos et.al [3]. -
Intrinsic Linking and Knotting of Graphs in Arbitrary 3–Manifolds 1 Introduction
Algebraic & Geometric Topology 6 (2006) 1025–1035 1025 arXiv version: fonts, pagination and layout may vary from AGT published version Intrinsic linking and knotting of graphs in arbitrary 3–manifolds ERICA FLAPAN HUGH HOWARDS DON LAWRENCE BLAKE MELLOR We prove that a graph is intrinsically linked in an arbitrary 3–manifold M if and only if it is intrinsically linked in S3 . Also, assuming the Poincare´ Conjecture, we prove that a graph is intrinsically knotted in M if and only if it is intrinsically knotted in S3 . 05C10, 57M25 1 Introduction The study of intrinsic linking and knotting began in 1983 when Conway and Gordon [1] 3 showed that every embedding of K6 (the complete graph on six vertices) in S contains 3 a non-trivial link, and every embedding of K7 in S contains a non-trivial knot. Since the existence of such a non-trivial link or knot depends only on the graph and not on the 3 particular embedding of the graph in S , we say that K6 is intrinsically linked and K7 is intrinsically knotted. At roughly the same time as Conway and Gordon’s result, Sachs [12, 11] independently proved that K6 and K3;3;1 are intrinsically linked, and used these two results to prove that any graph with a minor in the Petersen family (Figure 1) is intrinsically linked. Conversely, Sachs conjectured that any graph which is intrinsically linked contains a minor in the Petersen family. In 1995, Robertson, Seymour and Thomas [10] proved Sachs’ conjecture, and thus completely classified intrinsically linked graphs. -
On a Conjecture Concerning the Petersen Graph
On a conjecture concerning the Petersen graph Donald Nelson Michael D. Plummer Department of Mathematical Sciences Department of Mathematics Middle Tennessee State University Vanderbilt University Murfreesboro,TN37132,USA Nashville,TN37240,USA [email protected] [email protected] Neil Robertson* Xiaoya Zha† Department of Mathematics Department of Mathematical Sciences Ohio State University Middle Tennessee State University Columbus,OH43210,USA Murfreesboro,TN37132,USA [email protected] [email protected] Submitted: Oct 4, 2010; Accepted: Jan 10, 2011; Published: Jan 19, 2011 Mathematics Subject Classifications: 05C38, 05C40, 05C75 Abstract Robertson has conjectured that the only 3-connected, internally 4-con- nected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover, this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumption of internal 4-connectivity cannot be dropped as an hypothesis in the original conjecture. We then summarize our results aimed toward the solution of the conjec- ture in its original form. In particular, let G be any 3-connected internally-4- connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord. If C is any girth cycle in G then N(C)\V (C) cannot be edgeless, and if N(C)\V (C) contains a path of length at least 2, then the conjecture is true. Consequently, if the conjecture is false and H is a counterexample, then for any girth cycle C in H, N(C)\V (C) induces a nontrivial matching M together with an independent set of vertices. -
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). -
Knots, Links, Spatial Graphs, and Algebraic Invariants
689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society 689 Knots, Links, Spatial Graphs, and Algebraic Invariants AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory AMS Special Session on Spatial Graphs October 24–25, 2015 California State University, Fullerton, CA Erica Flapan Allison Henrich Aaron Kaestner Sam Nelson Editors American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Catherine Yan 2010 Mathematics Subject Classification. Primary 05C10, 57M15, 57M25, 57M27. Library of Congress Cataloging-in-Publication Data Names: Flapan, Erica, 1956- editor. Title: Knots, links, spatial graphs, and algebraic invariants : AMS special session on algebraic and combinatorial structures in knot theory, October 24-25, 2015, California State University, Fullerton, CA : AMS special session on spatial graphs, October 24-25, 2015, California State University, Fullerton, CA / Erica Flapan [and three others], editors. Description: Providence, Rhode Island : American Mathematical Society, [2017] | Series: Con- temporary mathematics ; volume 689 | Includes bibliographical references. Identifiers: LCCN 2016042011 | ISBN 9781470428471 (alk. paper) Subjects: LCSH: Knot theory–Congresses. | Link theory–Congresses. | Graph theory–Congresses. | Invariants–Congresses. | AMS: Combinatorics – Graph theory – Planar graphs; geometric and topological aspects of graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Relations with graph theory. msc | Manifolds and cell complexes – Low-dimensional topology – Knots and links in S3.msc| Manifolds and cell complexes – Low-dimensional topology – Invariants of knots and 3-manifolds. -
Snarks and Flow-Critical Graphs 1
Snarks and Flow-Critical Graphs 1 CˆandidaNunes da Silva a Lissa Pesci a Cl´audioL. Lucchesi b a DComp – CCTS – ufscar – Sorocaba, sp, Brazil b Faculty of Computing – facom-ufms – Campo Grande, ms, Brazil Abstract It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutte’s Flow Conjectures. Keywords: nowhere-zero k-flows, Tutte’s Flow Conjectures, 3-edge-colouring, flow-critical graphs. 1 Nowhere-Zero Flows Let k > 1 be an integer, let G be a graph, let D be an orientation of G and let ϕ be a weight function that associates to each edge of G a positive integer in the set {1, 2, . , k − 1}. The pair (D, ϕ) is a (nowhere-zero) k-flow of G 1 Support by fapesp, capes and cnpq if every vertex v of G is balanced, i. e., the sum of the weights of all edges leaving v equals the sum of the weights of all edges entering v. -
Every Generalized Petersen Graph Has a Tait Coloring
Pacific Journal of Mathematics EVERY GENERALIZED PETERSEN GRAPH HAS A TAIT COLORING FRANK CASTAGNA AND GEERT CALEB ERNST PRINS Vol. 40, No. 1 September 1972 PACIFIC JOURNAL OF MATHEMATICS Vol. 40, No. 1, 1972 EVERY GENERALIZED PETERSEN GRAPH HAS A TAIT COLORING FRANK CASTAGNA AND GEERT PRINS Watkins has defined a family of graphs which he calls generalized Petersen graphs. He conjectures that all but the original Petersen graph have a Tait coloring, and proves the conjecture for a large number of these graphs. In this paper it is shown that the conjecture is indeed true. DEFINITIONS. Let n and k be positive integers, k ^ n — 1, n Φ 2k. The generalized Petersen graph G(n, k) has 2n vertices, denoted by {0, 1, 2, , n - 1; 0', Γ, 2', , , (n - 1)'} and all edges of the form (ΐ, ί + 1), (i, i'), (ί', (i + k)f) for 0 ^ i ^ n — 1, where all numbers are read modulo n. G(5, 2) is the Petersen graph. See Watkins [2]. The sets of edges {(i, i + 1)} and {(i', (i + k)f)} are called the outer and inner rims respectively and the edges (£, i') are called the spokes. A Tait coloring of a trivalent graph is an edge-coloring in three colors such that each color is incident to each vertex. A 2-factor of a graph is a bivalent spanning subgraph. A 2-factor consists of dis- joint circuits. A Tait cycle of a trivalent graph is a 2~factor all of whose circuits have even length. A Tait cycle induces a Tait coloring and conversely. -
Lombardi Drawings of Graphs 1 Introduction
Lombardi Drawings of Graphs Christian A. Duncan1, David Eppstein2, Michael T. Goodrich2, Stephen G. Kobourov3, and Martin Nollenburg¨ 2 1Department of Computer Science, Louisiana Tech. Univ., Ruston, Louisiana, USA 2Department of Computer Science, University of California, Irvine, California, USA 3Department of Computer Science, University of Arizona, Tucson, Arizona, USA Abstract. We introduce the notion of Lombardi graph drawings, named after the American abstract artist Mark Lombardi. In these drawings, edges are represented as circular arcs rather than as line segments or polylines, and the vertices have perfect angular resolution: the edges are equally spaced around each vertex. We describe algorithms for finding Lombardi drawings of regular graphs, graphs of bounded degeneracy, and certain families of planar graphs. 1 Introduction The American artist Mark Lombardi [24] was famous for his drawings of social net- works representing conspiracy theories. Lombardi used curved arcs to represent edges, leading to a strong aesthetic quality and high readability. Inspired by this work, we intro- duce the notion of a Lombardi drawing of a graph, in which edges are drawn as circular arcs with perfect angular resolution: consecutive edges are evenly spaced around each vertex. While not all vertices have perfect angular resolution in Lombardi’s work, the even spacing of edges around vertices is clearly one of his aesthetic criteria; see Fig. 1. Traditional graph drawing methods rarely guarantee perfect angular resolution, but poor edge distribution can nevertheless lead to unreadable drawings. Additionally, while some tools provide options to draw edges as curves, most rely on straight-line edges, and it is known that maintaining good angular resolution can result in exponential draw- ing area for straight-line drawings of planar graphs [17,25]. -
Simple Treewidth Kolja Knauer 1 1 Introduction
Simple Treewidth Kolja Knauer 1 Technical University Berlin [email protected] Joint work with Torsten Ueckerdt 2. 1 Introduction A k-tree is a graph that can be constructed starting with a (k+1)-clique and in every step attaching a new vertex to a k-clique of the already constructed graph. The treewidth tw(G) of a graph G is the minimum k such that G is a partial k-tree, i.e., G is a subgraph of some k-tree [7]. We consider a variation of treewidth, called simple treewidth. A simple k-tree is a k-tree with the extra requirement that there is a construction sequence in which no two vertices are attached to the same k-clique. (Equiv- alently, a k-tree is simple if it has a tree representation of width k in which every (k − 1)-set of subtrees intersects at most 2 tree-vertices.) Now, the simple treewidth stw(G) of G is the minimum k such that G is a partial simple k-tree, i.e., G is a subgraph of some simple k-tree. We have encountered simple treewidth as a natural parameter in ques- tions concerning geometric representations of graphs, i.e., representing graphs as intersection graphs of geometrical objects where the quality of the repre- sentation is measured by the complexity of the objects. E.g., we have shown that the maximal interval-number([3]) of the class of treewidth k graphs is k + 1, whereas for the class of simple treewidth k graphs it is k, see [6]. -
When Graph Theory Meets Knot Theory
Contemporary Mathematics When graph theory meets knot theory Joel S. Foisy and Lewis D. Ludwig Abstract. Since the early 1980s, graph theory has been a favorite topic for undergraduate research due to its accessibility and breadth of applications. By the early 1990s, knot theory was recognized as another such area of mathe- matics, in large part due to C. Adams' text, The Knot Book. In this paper, we discuss the intersection of these two fields and provide a survey of current work in this area, much of which involved undergraduates. We will present several new directions one could consider for undergraduate work or one's own work. 1. Introduction This survey considers three current areas of study that combine the fields of graph theory and knot theory. Recall that a graph consists of a set of vertices and a set of edges that connect them. A spatial embedding of a graph is, informally, a way to place the graph in space. Historically, mathematicians have studied various graph embedding problems, such as classifying what graphs can be embedded in the plane (which is nicely stated in Kuratowski's Theorem [25]), and for non-planar graphs, what is the fewest number of crossings in a planar drawing (which is a difficult question for general graphs and still the subject of ongoing research, see [23] for example). A fairly recent development has been the investigation of graphs that have non-trivial links and knots in every spatial embedding. We say that a graph is intrinsically linked if it contains a pair of cycles that form a non-splittable link in every spatial embedding. -
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:1709.06469V2 [Math.CO]
ON DIHEDRAL FLOWS IN EMBEDDED GRAPHS Bart Litjens∗ Abstract. Let Γ be a multigraph with for each vertex a cyclic order of the edges incident with it. ±1 a For n 3, let D2n be the dihedral group of order 2n. Define D 0 1 a Z . Goodall, Krajewski, ≥ ∶= {( ) S ∈ } Regts and Vena in 2016 asked whether Γ admits a nowhere-identity D2n-flow if and only if it admits a nowhere-identity D-flow with a n (a ‘nowhere-identity dihedral n-flow’). We give counterexam- S S < ples to this statement and provide general obstructions. Furthermore, the complexity of deciding the existence of nowhere-identity 2-flows is discussed. Lastly, graphs in which the equivalence of the existence of flows as above is true, are described. We focus particularly on cubic graphs. Key words: embedded graph, cubic graph, dihedral group, flow, nonabelian flow MSC 2010: 05C10, 05C15, 05C21, 20B35 1 Introduction Let Γ V, E be a multigraph and G an additive abelian group. A nowhere-zero G-flow on Γ is an= assignment( ) of non-zero group elements to edges of Γ such that, for some orientation of the edges, Kirchhoff’s law is satisfied at every vertex. The following existence theorem is due to Tutte [16]. Theorem 1.1. Let Γ be a multigraph and n N. The following are equivalent statements. ∈ 1. There exists a nowhere-zero G-flow on Γ, for each abelian group G of order n. 2. There exists a nowhere-zero G-flow on Γ, for some abelian group G of order n.