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Two Poset Polytopes
Discrete Comput Geom 1:9-23 (1986) G eometrv)i.~.reh, ~ ( :*mllmlati~ml © l~fi $1~ter-Vtrlq New Yorklu¢. t¢ Two Poset Polytopes Richard P. Stanley* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract. Two convex polytopes, called the order polytope d)(P) and chain polytope <~(P), are associated with a finite poset P. There is a close interplay between the combinatorial structure of P and the geometric structure of E~(P). For instance, the order polynomial fl(P, m) of P and Ehrhart poly- nomial i(~9(P),m) of O(P) are related by f~(P,m+l)=i(d)(P),m). A "transfer map" then allows us to transfer properties of O(P) to W(P). In particular, we transfer known inequalities involving linear extensions of P to some new inequalities. I. The Order Polytope Our aim is to investigate two convex polytopes associated with a finite partially ordered set (poset) P. The first of these, which we call the "order polytope" and denote by O(P), has been the subject of considerable scrutiny, both explicit and implicit, Much of what we say about the order polytope will be essentially a review of well-known results, albeit ones scattered throughout the literature, sometimes in a rather obscure form. The second polytope, called the "chain polytope" and denoted if(P), seems never to have been previously considered per se. It is a special case of the vertex-packing polytope of a graph (see Section 2) but has many special properties not in general valid or meaningful for graphs. -
The AVD-Edge-Coloring Conjecture for Some Split Graphs
Matem´atica Contempor^anea, Vol. 44,1{10 c 2015, Sociedade Brasileira de Matem´atica The AVD-edge-coloring conjecture for some split graphs Alo´ısiode Menezes Vilas-B^oas C´eliaPicinin de Mello Abstract Let G be a simple graph. An adjacent vertex distinguishing edge- coloring (AVD-edge-coloring) of G is an edge-coloring of G such that for each pair of adjacent vertices u; v of G, the set of colors assigned to the edges incident with u differs from the set of colors assigned to the edges incident with v. The adjacent vertex distinguishing 0 chromatic index of G, denoted χa(G), is the minimum number of colors required to produce an AVD-edge-coloring for G. The AVD- edge-coloring conjecture states that every simple connected graph G ∼ 0 with at least three vertices and G =6 C5 has χa(G) ≤ ∆(G) + 2. The conjecture is open for arbitrary graphs, but it holds for some classes of graphs. In this note we focus on split graphs. We prove this AVD-edge- coloring conjecture for split-complete graphs and split-indifference graphs. 1 Introduction In this paper, G denotes a simple, undirected, finite, connected graph. The sets V (G) and E(G) are the vertex and edge sets of G. Let u; v 2 2000 AMS Subject Classification: 05C15. Key Words and Phrases: edge-coloring, adjacent strong edge-coloring, split graph. Supported by CNPq (132194/2010-4 and 308314/2013-1). The AVD-edge-coloring conjecture for some split graphs 2 V (G). We denote an edge by uv.A clique is a set of vertices pairwise adjacent in G and a stable set is a set of vertices such that no two of which are adjacent. -
JMM 2017 Student Poster Session Abstract Book
Abstracts for the MAA Undergraduate Poster Session Atlanta, GA January 6, 2017 Organized by Eric Ruggieri College of the Holy Cross and Chasen Smith Georgia Southern University Organized by the MAA Committee on Undergraduate Student Activities and Chapters and CUPM Subcommittee on Research by Undergraduates Dear Students, Advisors, Judges and Colleagues, If you look around today you will see over 300 posters and nearly 500 student presenters, representing a wide array of mathematical topics and ideas. These posters showcase the vibrant research being conducted as part of summer programs and during the academic year at colleges and universities from across the United States and beyond. It is so rewarding to see this session, which offers such a great opportunity for interaction between students and professional mathematicians, continue to grow. The judges you see here today are professional mathematicians from institutions around the world. They are advisors, colleagues, new Ph.D.s, and administrators. We have acknowledged many of them in this booklet; however, many judges here volunteered on site. Their support is vital to the success of the session and we thank them. We are supported financially by Tudor Investments and Two Sigma. We are also helped by the mem- bers of the Committee on Undergraduate Student Activities and Chapters (CUSAC) in some way or other. They are: Dora C. Ahmadi; Jennifer Bergner; Benjamin Galluzzo; Kristina Cole Garrett; TJ Hitchman; Cynthia Huffman; Aihua Li; Sara Louise Malec; Lisa Marano; May Mei; Stacy Ann Muir; Andy Nieder- maier; Pamela A. Richardson; Jennifer Schaefer; Peri Shereen; Eve Torrence; Violetta Vasilevska; Gerard A. -
Graph Theory, Prentice -Hall of India
Annexure 145 www.ccsenet.org/jmr Journal of Mathematics Research Vol. 3, No. 3; August 2011 Product Cordial Labeling in the Context of Tensor Product of Graphs S K Vaidya (Corresponding author) Department of Mathematics, Saurashtra University Rajkot-360 005. GUJARAT, India E-mail: [email protected] N B Vyas Atmiya Institute of Technology and Science Rajkot-360 005. GUJARAT, India E-mail: [email protected] Received: March 10, 2011 Accepted: March 28, 2011 doi:10.5539/jmr.v3n3p83 Abstract For the graph G1 and G2 the tensor product is denoted by G1(T p)G2 which is the graph with vertex set V(G1(T p)G2) = V(G1) × V(G2) and edge set E(G1(T p)G2) = {(u1, v1), (u2, v2)/u1u2E(G1) and v1v2E(G2)}. The graph Pm(T p)Pn is disconnected for ∀m, n while the graphs Cm(T p)Cn and Cm(T p)Pn are disconnected for both m and n even. We prove that these graphs are product cordial graphs. In addition to this we show that the graphs obtained by joining the connected components of respective graphs by a path of arbitrary length also admit product cordial labeling. Keywords: Cordial labeling, Porduct cordial labeling, Tensor product AMS Subject classification (2010): 05C78. 1. Introduction We begin with simple, finite and undirected graph G = (V(G), E(G)). For standard terminology and notations we follow (West, D. B, 2001). The brief summary of definitions and relevant results are given below. 1.1 Definition: If the vertices of the graph are assigned values subject to certain condition(s) then it is known as graph labeling. -
Double Posets and Real Invariant Varieties Dissertation
Fachbereich Mathematik und Informatik der Freien Universität Berlin Double posets and real invariant varieties Two interactions between combinatorics and geometry Dissertation eingereicht von Tobias Friedl Berlin 2017 Advisor and first reviewer: Prof. Dr. Raman Sanyal Second reviewer: Prof. Dr. Francisco Santos Third reviewer: Priv.-Doz. Dr. Christian Stump Date of the defense: May 19, 2017 Acknowledgements My deepest thanks go to my advisor Raman Sanyal. A PhD-student can only hope for an advisor who is as dedicated and enthusiastic about mathematics as you are. Thank you for getting your hands dirty and spending many hours in front of the blackboard teaching me how to do research. I spent the last years in the amazing and inspiring work group directed by Günter Ziegler at FU Berlin. Thank you for providing such a welcoming and challenging work environment. I really enjoyed the time with my friends and colleagues at the "villa", most of all Francesco Grande, Katharina Jochemko, Katy Beeler, Albert Haase, Lauri Loiskekoski, Philip Brinkmann, Nevena Palić and Jean-Philippe Labbé. Thanks also to Elke Pose, who provides valuable support and keeps bureaucracy at a low level. I want to thank my coauthor Cordian Riener for many fruitful mathematical discussions and for an interesting and enjoyable week of research in Helsinki at Aalto University. Moreover, I’d like to express my gratitude towards my coauthor and friend Tom Chappell. Thanks to the reviewers Paco Santos and Christian Stump for their helpful com- ments and to Christian additionally for many discussions regarding the combinatorics of reflection groups and posets throughout the last years. -
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). -
Arxiv:1902.07301V3 [Math.CO] 13 Feb 2020 Which N H Itiuinwhere Distribution the and of Ojcuefo 8]Wihhsntbe Eovd H Oere Sole the Resolved
MINUSCULE DOPPELGANGERS,¨ THE COINCIDENTAL DOWN-DEGREE EXPECTATIONS PROPERTY, AND ROWMOTION SAM HOPKINS Abstract. We relate Reiner, Tenner, and Yong’s coincidental down-degree expec- tations (CDE) property of posets to the minuscule doppelg¨anger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelg¨anger pairs behave “as if” they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelg¨anger pairs pretending to have isomor- phic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelg¨anger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type). 1. Introduction Let P be a finite poset. The down-degree of p ∈ P is the number of elements of P which p covers. Consider two probability distributions on P : the uniform distribution; and the distribution where p ∈ P occurs proportional to the number of maximal chains of P containing p. We say that P has the coincidental down-degree expectations (CDE) property if the expected value of the down-degree statistic is the same for these two distributions. -
Restricted Colorings of Graphs
Restricted colorings of graphs Noga Alon ∗ Department of Mathematics Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Tel Aviv, Israel and Bellcore, Morristown, NJ 07960, USA Abstract The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well. ∗Research supported in part by a United States Israel BSF Grant. Appeared in "Surveys in Combinatorics", Proc. 14th British Combinatorial Conference, London Mathematical Society Lecture Notes Series 187, edited by K. Walker, Cambridge University Press, 1993, 1-33. 0 1 Introduction Graph coloring is arguably the most popular subject in graph theory. An interesting variant of the classical problem of coloring properly the vertices of a graph with the minimum possible number of colors arises when one imposes some restrictions on the colors available for every vertex. This variant received a considerable amount of attention that led to several fascinating conjectures and results, and its study combines interesting combinatorial techniques with powerful algebraic and probabilistic ideas. The subject, initiated independently by Vizing [51] and by Erd}os,Rubin and Taylor [24], is usually known as the study of the choosability properties of a graph. In the present paper we survey some of the known recent and less recent results in this topic, focusing on the techniques involved and mentioning some of the related intriguing open problems. -
Theoretical Computer Science a Polynomial Solution to the K-Fixed
Theoretical Computer Science 411 (2010) 967–975 Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: www.elsevier.com/locate/tcs A polynomial solution to the k-fixed-endpoint path cover problem on proper interval graphs Katerina Asdre, Stavros D. Nikolopoulos ∗ Department of Computer Science, University of Ioannina, P.O. Box 1186, GR-45110 Ioannina, Greece article info a b s t r a c t Article history: We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover Received 29 January 2008 problem, or kPC for short. Given a graph G and a subset T of k vertices of V .G/, a k-fixed- Received in revised form 24 December 2008 endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers Accepted 7 November 2009 the vertices of G such that the k vertices of T are all endpoints of the paths in P . The kPC Communicated by P. Spirakis problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T is empty (or, equivalently, k D 0), the stated problem coincides with the classical path Keywords: cover problem. The kPC problem generalizes some path cover related problems, such as Perfect graphs Proper interval graphs the 1HP and 2HP problems, which have been proved to be NP-complete. Note that the Path cover complexity status for both 1HP and 2HP problems on interval graphs remains an open Fixed-endpoint path cover question (Damaschke (1993) [9]). In this paper, we show that the kPC problem can be solved Linear-time algorithms in linear time on the class of proper interval graphs, that is, in O.n C m/ time on a proper interval graph on n vertices and m edges. -
The Algebra of Projective Spheres on Plane, Sphere and Hemisphere
Journal of Applied Mathematics and Physics, 2020, 8, 2286-2333 https://www.scirp.org/journal/jamp ISSN Online: 2327-4379 ISSN Print: 2327-4352 The Algebra of Projective Spheres on Plane, Sphere and Hemisphere István Lénárt Eötvös Loránd University, Budapest, Hungary How to cite this paper: Lénárt, I. (2020) Abstract The Algebra of Projective Spheres on Plane, Sphere and Hemisphere. Journal of Applied Numerous authors studied polarities in incidence structures or algebrization Mathematics and Physics, 8, 2286-2333. of projective geometry [1] [2]. The purpose of the present work is to establish https://doi.org/10.4236/jamp.2020.810171 an algebraic system based on elementary concepts of spherical geometry, ex- tended to hyperbolic and plane geometry. The guiding principle is: “The Received: July 17, 2020 Accepted: October 27, 2020 point and the straight line are one and the same”. Points and straight lines are Published: October 30, 2020 not treated as dual elements in two separate sets, but identical elements with- in a single set endowed with a binary operation and appropriate axioms. It Copyright © 2020 by author(s) and consists of three sections. In Section 1 I build an algebraic system based on Scientific Research Publishing Inc. This work is licensed under the Creative spherical constructions with two axioms: ab= ba and (ab)( ac) = a , pro- Commons Attribution International viding finite and infinite models and proving classical theorems that are License (CC BY 4.0). adapted to the new system. In Section Two I arrange hyperbolic points and http://creativecommons.org/licenses/by/4.0/ straight lines into a model of a projective sphere, show the connection be- Open Access tween the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. -
An Upper Bound for Cubicity in Terms of Boxicity
An upper bound for Cubicity in terms of Boxicity L. Sunil Chandran ∗ K. Ashik Mathew † Abstract An axis-parallel b-dimensional box is a Cartesian product R1 × R2 × ... × Rb where each Ri (for 1 ≤ i ≤ b) is a closed interval of the form [ai,bi] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product R1 × R2 × ... × Rb, where each Ri (for 1 ≤ i ≤ b) is a closed interval of the form [ai,ai+1] on the real line. When the boxes are restricted to be axis-parallel cubes in b-dimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by cub(G)). In this paper we prove that cub(G) ≤⌈log n⌉box(G) where n is the number of vertices in the graph. We also show that this upper bound is tight. Keywords: Cubicity, Boxicity, Interval graph, Indifference graph 1 Introduction Let F = {Sx ⊆ U : x ∈ V } be a family of subsets of U, where V is an index set. The intersec- tion graph Ω(F) of F has V as vertex set, and two distinct vertices x and y are adjacent if and only if Sx ∩ Sy 6= ∅. Representations of graphs as the intersection graphs of various geometric objects is a well-studied area in graph theory. A prime example of a graph class defined in this way is the class of interval graphs. -
On Pancyclism in Hamiltonian Graphs
CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Discrete Mathematics 251 (2002) 119–127 www.elsevier.com/locate/disc On pancyclism in hamiltonian graphs Mekkia Kouidera, Antoni Marczykb; ∗ aL.R.I, Universiteà de Paris-Sud, Bat.ˆ 490, 91405 Orsay, France bFaculty of Applied Mathematics, AGH, Al. Mickiewicza 30, 30-059 Krakow,à Poland Received 11 May 1999; revised 5 November 1999; accepted 3 June 2001 Abstract We investigate the set of cycle lengths occurring in a hamiltonian graph with at least one or two vertices of large degree. We prove that in every case this set contains all the integers between 3 and some t, where t depends on the order of the graph and the degrees of vertices. c 2002 Elsevier Science B.V. All rights reserved. MSC: 05C38; 05C45 Keywords: Hamiltonian graphs; Pancyclic graphs; Cycles 1. Introduction Throughout this paper, we consider only ÿnite, undirected and simple graphs. For a graph G we denote by V = V (G) its vertex set, by E = E(G) its set of edges. By Cp we mean a p-cycle of G, i.e., a cycle of length p. The vertices of a graph G of order n will be denoted by integers 0; 1; 2;:::;n− 1 and considered modulo n.We write EG(k)=E(k) for the set of edges that are incident to a vertex k. For a subgraph H of G and a vertex k of G; dH (k) denotes the number of all edges (k; j)ofE(k) with j ∈ V (H), i.e., if k ∈ V (H) then dH (k) is the degree of k in the subgraph of G induced by the set V (H).