Lombardi Drawings of Graphs 1 Introduction
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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]. -
Automorphisms of the Double Cover of a Circulant Graph of Valency at Most 7
AUTOMORPHISMS OF THE DOUBLE COVER OF A CIRCULANT GRAPH OF VALENCY AT MOST 7 ADEMIR HUJDUROVIC,´ ¯DOR¯DE MITROVIC,´ AND DAVE WITTE MORRIS Abstract. A graph X is said to be unstable if the direct product X × K2 (also called the canonical double cover of X) has automorphisms that do not come from automorphisms of its factors X and K2. It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors. We find all of the nontrivially unstable circulant graphs of valency at most 7. (They come in several infinite families.) We also show that the instability of each of these graphs is explained by theorems of Steve Wilson. This is best possible, because there is a nontrivially unstable circulant graph of valency 8 that does not satisfy the hypotheses of any of Wilson's four instability theorems for circulant graphs. 1. Introduction Let X be a circulant graph. (All graphs in this paper are finite, simple, and undirected.) Definition 1.1 ([16]). The canonical bipartite double cover of X is the bipartite graph BX with V (BX) = V (X) 0; 1 , where × f g (v; 0) is adjacent to (w; 1) in BX v is adjacent to w in X: () Letting S2 be the symmetric group on the 2-element set 0; 1 , it is clear f g that the direct product Aut X S2 is a subgroup of Aut BX. We are interested in cases where this subgroup× is proper: Definition 1.2 ([12, p. -
Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-Connected Graphs with Bounded Maximum Degrees
Georgia State University ScholarWorks @ Georgia State University Mathematics Dissertations Department of Mathematics and Statistics 5-11-2015 Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees Songling Shan Follow this and additional works at: https://scholarworks.gsu.edu/math_diss Recommended Citation Shan, Songling, "Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees." Dissertation, Georgia State University, 2015. https://scholarworks.gsu.edu/math_diss/23 This Dissertation is brought to you for free and open access by the Department of Mathematics and Statistics at ScholarWorks @ Georgia State University. It has been accepted for inclusion in Mathematics Dissertations by an authorized administrator of ScholarWorks @ Georgia State University. For more information, please contact [email protected]. Homeomorphically Irreducible Spanning Trees, Halin Graphs, and Long Cycles in 3-connected Graphs with Bounded Maximum Degrees by Songling Shan Under the Direction of Guantao Chen, PhD ABSTRACT A tree T with no vertex of degree 2 is called a homeomorphically irreducible tree (HIT) and if T is spanning in a graph, then T is called a homeomorphically irreducible spanning tree (HIST). Albertson, Berman, Hutchinson and Thomassen asked if every triangulation of at least 4 vertices has a HIST and if every connected graph with each edge in at least two triangles contains a HIST. These two questions were restated as two conjectures by Archdeacon in 2009. The first part of this dissertation gives a proof for each of the two conjectures. The second part focuses on some problems about Halin graphs, which is a class of graphs closely related to HITs and HISTs. -
Drawing Graphs and Maps with Curves
Report from Dagstuhl Seminar 13151 Drawing Graphs and Maps with Curves Edited by Stephen Kobourov1, Martin Nöllenburg2, and Monique Teillaud3 1 University of Arizona – Tucson, US, [email protected] 2 KIT – Karlsruhe Institute of Technology, DE, [email protected] 3 INRIA Sophia Antipolis – Méditerranée, FR, [email protected] Abstract This report documents the program and the outcomes of Dagstuhl Seminar 13151 “Drawing Graphs and Maps with Curves”. The seminar brought together 34 researchers from different areas such as graph drawing, information visualization, computational geometry, and cartography. During the seminar we started with seven overview talks on the use of curves in the different communities represented in the seminar. Abstracts of these talks are collected in this report. Six working groups formed around open research problems related to the seminar topic and we report about their findings. Finally, the seminar was accompanied by the art exhibition Bending Reality: Where Arc and Science Meet with 40 exhibits contributed by the seminar participants. Seminar 07.–12. April, 2013 – www.dagstuhl.de/13151 1998 ACM Subject Classification I.3.5 Computational Geometry and Object Modeling, G.2.2 Graph Theory, F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases graph drawing, information visualization, computational cartography, computational geometry Digital Object Identifier 10.4230/DagRep.3.4.34 Edited in cooperation with Benjamin Niedermann 1 Executive Summary Stephen Kobourov Martin Nöllenburg Monique Teillaud License Creative Commons BY 3.0 Unported license © Stephen Kobourov, Martin Nöllenburg, and Monique Teillaud Graphs and networks, maps and schematic map representations are frequently used in many fields of science, humanities and the arts. -
Graph Operations and Upper Bounds on Graph Homomorphism Counts
Graph Operations and Upper Bounds on Graph Homomorphism Counts Luke Sernau March 9, 2017 Abstract We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any n-vertex, d-regular graph G and any graph H (possibly with loops), n n d d hom(G, H) ≤ max hom(Kd,d,H) 2 , hom(Kd+1,H) +1 , n o where hom(G, H) is the number of homomorphisms from G to H. By exploiting properties of the graph tensor product and graph exponentiation, we also find new infinite families of H for which the bound stated above on hom(G, H) holds for all n-vertex, d-regular G. In particular we show that if HWR is the complete looped path on three vertices, also known as the Widom-Rowlinson graph, then n d hom(G, HWR) ≤ hom(Kd+1,HWR) +1 for all n-vertex, d-regular G. This verifies a conjecture of Galvin. arXiv:1510.01833v3 [math.CO] 8 Mar 2017 1 Introduction Graph homomorphisms are an important concept in many areas of graph theory. A graph homomorphism is simply an adjacency-preserving map be- tween a graph G and a graph H. That is, for given graphs G and H, a function φ : V (G) → V (H) is said to be a homomorphism from G to H if for every edge uv ∈ E(G), we have φ(u)φ(v) ∈ E(H) (Here, as throughout, all graphs are simple, meaning without multi-edges, but they are permitted 1 to have loops). -
Zadání Bakalářské Práce
ČESKÉ VYSOKÉ UČENÍ TECHNICKÉ V PRAZE FAKULTA INFORMAČNÍCH TECHNOLOGIÍ ZADÁNÍ BAKALÁŘSKÉ PRÁCE Název: Rekurzivně konstruovatelné grafy Student: Anežka Štěpánková Vedoucí: RNDr. Jiřina Scholtzová, Ph.D. Studijní program: Informatika Studijní obor: Teoretická informatika Katedra: Katedra teoretické informatiky Platnost zadání: Do konce letního semestru 2016/17 Pokyny pro vypracování Souvislé rekurentně zadané grafy jsou souvislé grafy, pro které existuje postup, jak ze zadaného grafu odvodit graf nový. V předmětu BI-GRA jste se s jednoduchými rekurentními grafy setkali, vždy šly popsat lineárními rekurentními rovnicemi s konstantními koeficienty. Existuje klasifikace, která definuje třídu rekurzivně konstruovatelných grafů [1]. Podobně jako u grafů v BI-GRA zjistěte, zda lze některé z kategorií rekurzivně konstruovatelných grafů popsat rekurentními rovnicemi. 1. Seznamte se s rekurentně zadanými grafy. 2. Seznamte se s rekurzivně konstruovatelnými grafy a nastudujte články o nich. 3. Vyberte si některé z kategorií rekurzivně konstruovatelných grafů [1]. 4. Pro vybrané kategorie najděte způsob, jak je popsat využitím rekurentních rovnic. Seznam odborné literatury [1] Gross, Jonathan L., Jay Yellen, Ping Zhang. Handbook of Graph Theory, 2nd ed, Boca Raton: CRC, 2013. ISBN: 978-1-4398-8018-0 doc. Ing. Jan Janoušek, Ph.D. prof. Ing. Pavel Tvrdík, CSc. vedoucí katedry děkan V Praze dne 18. února 2016 České vysoké učení technické v Praze Fakulta informačních technologií Katedra Teoretické informatiky Bakalářská práce Rekurzivně konstruovatelné grafy Anežka Štěpánková Vedoucí práce: RNDr. Jiřina Scholtzová Ph.D. 12. května 2017 Poděkování Děkuji vedoucí práce RNDr. Jiřině Scholtzové Ph.D., která mi ochotně vyšla vstříc s volbou tématu a v průběhu tvorby práce vždy dobře poradila a na- směrovala mne správným směrem. -
Vertex-Transitive Graphs That Have No Hamilton Decomposition
Vertex-transitive graphs that have no Hamilton decomposition Darryn Bryant ∗ Matthew Dean † Abstract It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency. 1 Introduction A famous question of Lov´asz concerns the existence of Hamilton paths in vertex-transitive graphs [28], and no example of a connected vertex-transitive graph with no Hamilton path is known. The related question concerning the existence of Hamilton cycles in vertex-transitive graphs is another interesting and well-studied problem in graph theory, see the survey [23]. A Hamiltonian graph is a graph containing a Hamilton cycle. Thomassen (see [10, 23]) has conjectured that there are only finitely many non-Hamiltonian connected vertex-transitive graphs. On the other hand, Babai [8, 9] has conjectured that there are infinitely many such graphs. To date only five are known. These arXiv:1408.5211v3 [math.CO] 12 Nov 2014 are the complete graph of order 2, the Petersen graph, the Coxeter graph, and the two graphs obtained from the Petersen and Coxeter graphs by replacing each vertex with a triangle. For a regular graph of valency at least 4, a stronger property than the existence of a Hamilton cycle is the existence of a Hamilton decomposition. If X is a k-valent graph, then a Hamilton k decomposition of X is a set of ⌊ 2 ⌋ pairwise edge-disjoint Hamilton cycles in X. Given the small number of non-Hamiltonian connected vertex-transitive graphs, and the uncertainty concerning the existence of others, it is natural to ask how many connected vertex-transitive graphs have no Hamilton decomposition. -
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). -
A Survey of Graph Coloring - Its Types, Methods and Applications
FOUNDATIONS OF COMPUTING AND DECISION SCIENCES Vol. 37 (2012) No. 3 DOI: 10.2478/v10209-011-0012-y A SURVEY OF GRAPH COLORING - ITS TYPES, METHODS AND APPLICATIONS Piotr FORMANOWICZ1;2, Krzysztof TANA1 Abstract. Graph coloring is one of the best known, popular and extensively researched subject in the eld of graph theory, having many applications and con- jectures, which are still open and studied by various mathematicians and computer scientists along the world. In this paper we present a survey of graph coloring as an important subeld of graph theory, describing various methods of the coloring, and a list of problems and conjectures associated with them. Lastly, we turn our attention to cubic graphs, a class of graphs, which has been found to be very interesting to study and color. A brief review of graph coloring methods (in Polish) was given by Kubale in [32] and a more detailed one in a book by the same author. We extend this review and explore the eld of graph coloring further, describing various results obtained by other authors and show some interesting applications of this eld of graph theory. Keywords: graph coloring, vertex coloring, edge coloring, complexity, algorithms 1 Introduction Graph coloring is one of the most important, well-known and studied subelds of graph theory. An evidence of this can be found in various papers and books, in which the coloring is studied, and the problems and conjectures associated with this eld of research are being described and solved. Good examples of such works are [27] and [28]. In the following sections of this paper, we describe a brief history of graph coloring and give a tour through types of coloring, problems and conjectures associated with them, and applications. -
Minor-Closed Graph Classes with Bounded Layered Pathwidth
Minor-Closed Graph Classes with Bounded Layered Pathwidth Vida Dujmovi´c z David Eppstein y Gwena¨elJoret x Pat Morin ∗ David R. Wood { 19th October 2018; revised 4th June 2020 Abstract We prove that a minor-closed class of graphs has bounded layered pathwidth if and only if some apex-forest is not in the class. This generalises a theorem of Robertson and Seymour, which says that a minor-closed class of graphs has bounded pathwidth if and only if some forest is not in the class. 1 Introduction Pathwidth and treewidth are graph parameters that respectively measure how similar a given graph is to a path or a tree. These parameters are of fundamental importance in structural graph theory, especially in Roberston and Seymour's graph minors series. They also have numerous applications in algorithmic graph theory. Indeed, many NP-complete problems are solvable in polynomial time on graphs of bounded treewidth [23]. Recently, Dujmovi´c,Morin, and Wood [19] introduced the notion of layered treewidth. Loosely speaking, a graph has bounded layered treewidth if it has a tree decomposition and a layering such that each bag of the tree decomposition contains a bounded number of vertices in each layer (defined formally below). This definition is interesting since several natural graph classes, such as planar graphs, that have unbounded treewidth have bounded layered treewidth. Bannister, Devanny, Dujmovi´c,Eppstein, and Wood [1] introduced layered pathwidth, which is analogous to layered treewidth where the tree decomposition is arXiv:1810.08314v2 [math.CO] 4 Jun 2020 required to be a path decomposition. -
A Note on Max-Leaves Spanning Tree Problem in Halin Graphs
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 29 (2004), Pages 95–97 A note on max-leaves spanning tree problem in Halin graphs Dingjun Lou Huiquan Zhu Department of Computer Science Zhongshan University Guangzhou 510275 People’s Republic of China Abstract A Halin graph H is a planar graph obtained by drawing a tree T in the plane, where T has no vertex of degree 2, then drawing a cycle C through all leaves in the plane. We write H = T ∪ C,whereT is called the characteristic tree and C is called the accompanying cycle. The problem is to find a spanning tree with the maximum number of leaves in a Halin graph. In this paper, we prove that the characteristic tree in a Halin graph H is one of the spanning trees with maximum number of leaves in H, and we design a linear time algorithm to find such a tree in the Halin graph. All graphs in this paper are finite, undirected and simple. We follow the termi- nology and notation of [1] and [6]. A Halin graph H is a planar graph obtained by drawing a tree T in the plane, where T has no vertex of degree 2, then drawing a cycle through all leaves of T in the plane. We write H = T ∪ C,whereT is called the characteristic tree and C is called the accompanying cycle. In this paper, we deal with the problem of finding a spanning tree with maximum number of leaves in a Halin graph. The problem of finding a spanning tree with maximum number of leaves for arbitrary graphs is NP-complete (see [3], Problem [ND 2]). -
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.