DOCSLIB.ORG

Math 235 Homefun Exercises

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Math 235 Homefun Exercises Math 235 Homefun Exercises Dr. G Fall 2017 1: WTF is Discrete Math? It’s time to get you set up with LATEX, a typesetting language that does a much better job of creating professional-looking documents than standard word processors like Microsoft Word. The King’s College Math and Computer Science departments both require their students to become proficient in LATEX. Now is the time to get started! Go to the computer lab, find and run the application TeXstudio, and open the document Welcome to LaTeX.tex that I emailed to you. Compile it by pressing the F5 key. Read the whole thing! 2: Graph Theory Fundamentals 1. Draw a graph with five vertices v1, v2, v3, v4, and v5 such that the degree of v1 is 3, v2 has odd degree, v3 has degree 2, and v4 and v5 are adjacent. 2. For all the graphs you’ve seen so far, go back and count the number of edges in each. Then, add up the degrees of all the vertices. You should see a connection between the number of edges and the sum of all degrees. State that connection clearly and succinctly. 3. If a vertex has an even degree (2, 4, 6, etc.), then we call it an even vertex. Likewise, a vertex having an odd degree is called an odd vertex. Use what you found in the previous exercise to prove that in any graph, the number of odd vertices must be even. 4. Consider a graph with seven vertices. Five vertices have degree 4 and two vertices have degree 2. How many edges does this graph have? 5. Consider a graph whose degree sequence is 5; 5; 4; 4; 3; 3; 3; 3. How many edges does this graph have? 6. What is the degree sequence of the complete graph Kn? 7. How many edges does the complete graph Kn have? Experiment for specific small values of n, and then give the general answer in terms of n. 8. Draw graphs having the following degree sequence, or else explain why no such graph can exist. 1 (a) 1, 1, 1, 1 (b) 1, 1, 1, 1, 1 (c) 2, 2, 2, 2, 2 (d) 4, 4, 3, 2, 2, 0 (e) 3, 3, 3, 3, 2, 2, 2 (f) 3, 3, 3, 3, 3, 3 (g) 6, 6, 4, 2, 2, 2, 1, 1 9. Try to draw a graph where the degrees of the vertices are all different. Can you do it? If so, show me. If not, try to prove why you believe it’s impossible. 10. Imagine the complete graph K6 where each edge has been colored either red or blue. Prove that this graph contains either a red triangle or a blue triangle. 11. Show that K5 can have its edges colored red and blue with no red nor blue triangles. 12. Suppose all vertices in a graph have the same odd degree k. Show that the number of edges is a multiple of k. 13. Show by example that if the vertices in a graph all have the same even degree k, the number of edges may or may not be a multiple of k. 3: Cycles 1.A cut vertex is any vertex in a connected graph that, if deleted, would disconnect the graph. (a) Draw a graph with 5 vertices, one of which is a cut vertex. (b) Draw a graph with 5 vertices such that every vertex is a cut vertex. (c) Explain why no graph having a cut vertex can be Hamiltonian. 2. Suppose that if G is a connected graph and C is a cycle within G. Prove that if one of the edges of C is deleted, G remains connected. 3. A graph G with n vertices is called pancyclic if it contains cycles of every length 3, 4, ..., n. For each of n = 4, 5, and 6, draw a pancyclic graph with n vertices. To make it more interesting, use as few edges as possible. 4. Determine whether the following graphs are Hamiltonian: 5. Prove that any graph with degree sequence 4; 4; 4; 4; 4; 4 is Hamiltonian. 6. The wheel graph Wn consists of a cycle graph Cn together with one extra vertex that is adjacent to all other vertices (so actually, the wheel graph Wn has n + 1 vertices). For example, here is W6: (a) What is the degree sequence of Wn? (b) Prove that Wn is Hamiltonian for all n ≥ 3. 7. Solve the Traveling Salesperson Problem for the weighted graph below two ways: first using the Nearest Neighbor algorithm (start at the top vertex), then using Cheapest Link. 7 1 6 3 4 10 8 2 5 9 8. There are 8 different binary strings of length 3: 000; 100; 010; 001; 110; 101; 011; 111 Draw a graph where each vertex represents one of these strings. Draw an edge between two vertices if and only if the strings differ only in one digit. For example, the strings 100 and 110 differ only in the second digit, so you should draw an edge between these two vertices. Find a Hamilton cycle in this graph. 9. The girth of a graph is the length of its shortest cycle. What is the girth of the complete graph Kn? What is the girth of the cycle graph Cn? 10. Find the girth of the following graph: Note: this graph is famous. It’s called the Petersen Graph. You’ll be seeing it a lot. 4: Circuits 1. Use Fleury’s algorithm to find an Euler circuit in the graph below: 2. Find an Euler circuit in each of the graphs below: G A F B E C D 3. Consider the graph below: (a) Explain why this graph does not possess an Euler circuit. (b) Label the vertices, and then give an Euler path in this graph. 4. For which values of n does the complete graph Kn have an Euler circuit? 5.A bridge is any edge in a connected graph that, if deleted, would disconnect the graph. (a) Draw a graph with 6 vertices having exactly one bridge. (b) Draw a graph with 6 vertices such that every edge is a bridge. (c) Explain why no graph having a bridge can have an Euler circuit. (d) Show by examples that a graph having a bridge may or may not have an Euler path. 6. List all bridges in the following graph: A B G C G D F E Hint: This is actually kinda tricky because of the way the edges are crossing each other. If you try redrawing the graph without any edges crossing, finding the bridges becomes much easier. 7. We saw one practical application of Euler circuits: the Trash Collector Problem. Find at least two other “real world” situations that amount to finding an Euler circuit in some graph. 8. A graph is randomly traceable from a vertex v if, whenever we start from v and traverse the graph in an arbitrary way never retracing any edge, we eventually obtain an Euler circuit. (a) Show that the graph below is randomly traceable from the marked vertex v. v (b) Show that this graph is not randomly traceable from any other vertex. 5: Trees 1. If T is a tree with n vertices, what is the maximum degree any one vertex can have? 2. If T is a tree with n vertices, what is the maximum number of leaves (i.e. vertices of degree 1) T can have? 3. Prove that if T is a tree, then adding any new edge creates a cycle. 4. I’m very bad at chess, so I’ve decided that the next time I play, I will quit after winning a game or after five games have been played. Make a tree showing all possible outcomes. How many outcomes are there? In how many of these outcomes do I win a game? 5. It’s the weekend and I have several ways to spend my Friday and Saturday evenings. I could do some cleaning, watch cat videos on Youtube, or go out with my friends. Make a tree showing all the possible ways I can spend my Friday and Saturday evenings. 6. Draw some trees and count how many edges they have. Then finish this conjecture: If a tree has n vertices, then it has edges. Try to prove your conjecture. 7. Solve the minimal connector problem for this edge-weighted graph. What is the total weight of the spanning tree you found? 3 5 A B C 2 5 5 7 1 9 1 D E F 1 3 6 8 3 G H I 7 4 8. Application to Chemistry: One of the earliest uses of graphs was to enumerate chemical molecules. If we have a molecule consisting only of carbon and hydrogen atoms, then we can represent it as a graph in which each carbon atom appears as a vertex of degree 4 and each hydrogen atom appears as a vertex of degree 1. The graphs of n-butane and 2-methyl propane are shown below. Although they have the same chemical formula C4H10, they are different molecules because the atoms are arranged differently within the molecule. These two molecules form part of a general class of molecules known as the alkanes, or paraffins, with chemical formula CnH2n+2. It is natural to ask how many different molecules there are with this formula. H H C H H H H H H H H C C C C H H C C C H H H H H H H H To find all possible molecules with the formula CnH2n+2, note that the molecule is completely determined by how the carbon atoms are arranged; once the carbon atoms are set, the hydrogen atoms just fill in to bring the degree of each carbon vertex to 4.
Load more

Recommended publications
  • 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).
    [Show full text]
  • On the Pancyclicity of 1-Tough Graphs 1
    Discrete Mathematics Letters Discrete Math. Lett. 5 (2021) 1–4 www.dmlett.com DOI: 10.47443/dml.2020.0046 Research Article On the pancyclicity of 1-tough graphs Rao Li∗ Department of Mathematical Sciences, University of South Carolina Aiken, Aiken, SC 29801, USA (Received: 14 August 2020. Received in revised form: 5 October 2020. Accepted: 13 November 2020. Published online: 23 November 2020.) c 2020 the author. This is an open access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/) Abstract Let G be a graph of order n ≥ 6r−2, size e, and minimum degree δ ≥ r, where r is an integer greater than 1. The main result obtained in this note is that if G is 1-tough with the degree sequence (d1; d2; ··· ; dn) and if e ≥ ((n−r)(n−r−1)+r(2r−1))=2, then G is pancyclic, Hamiltonian bipartite, or Hamiltonian such that d1 = d2 = ··· = dk = k, dk+1 = dk+2 = ··· = dn−k+1 = n − k − 1, and dn−k+2 = dn−k+3 = ··· = dn = n − 1, where k < n=2. This result implies that the following conjecture of Hoa, posed in 2002, is true under the conditions n ≥ 40 and δ ≥ 7: every path-tough graph on n vertices and with at least ((n − 6)(n − 7) + 34)=2 edges is Hamiltonian. Using the main result of this note, additional sufficient conditions for 1-tough graphs to be pancyclic are also obtained. Keywords: pancyclic graph; 1-tough graph; Hamiltonian graph. 2020 Mathematics Subject Classification: 05C45.
    [Show full text]
  • A Neighborhood and Degree Condition for Pancyclicity and Vertex Pancyclicity
    AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 40 (2008), Pages 15–25 A neighborhood and degree condition for pancyclicity and vertex pancyclicity Ryota Matsubara Masao Tsugaki Department of Mathematical Information Science Tokyo University of Science 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601 Japan [email protected] [email protected] Tomoki Yamashita Department of Mathematics School of Dentistry, Asahi University 1851 Hozumi, Gifu 501–0296 Japan [email protected] Abstract Let G be a 2-connected graph of order n. For any u ∈ V (G) and l ∈ {m, m + 1,...,n}, if G has a cycle of length l, then G is called [m, n]- pancyclic, and if G has a cycle of length l which contains u, then G is called [m, n]-vertex pancyclic. Let δ(G) be a minimum degree of G and let NG(x) be the neighborhood of a vertex x in G. In [Australas. J. Combin. 12 (1995), 81–91] Liu, Lou and Zhao proved that if |NG(u) ∪ NG(v)| + δ(G) ≥ n+1 for any nonadjacent vertices u, v of G, then G is [3, n]-vertex pancyclic. In this paper, we prove if n ≥ 6 and |NG(u)∪NG(v)|+dG(w) ≥ n for every triple independent vertices u,v,w of G, then (i) G is [3, n]- pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2, and (ii) G is [5, n]-vertex pancyclic or isomorphic to the complete bipartite graph Kn/2,n/2. 1 Introduction In this paper, we consider only finite graphs without loops or multiple edges.
    [Show full text]
  • Vertex Pancyclic Graphs
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Applied Mathematics 120 (2002) 219–237 Vertex pancyclic graphs Bert Randeratha; ∗,Ingo Schiermeyer b,Meike Tewes b,Lutz Volkmann c aInstitut fur Informatik, Universitat zu Koln, Pohligstrasse 1 50969 Koln, Germany bFakultat fur Mathematik und Informatik, TU Bergakademie Freiberg, 09596 Freiberg, Germany cLehrstuhl II fur Mathematik, RWTH Aachen, 52056 Aachen, Germany Received 17 September 1999; received in revised form 31 July 2000; accepted 21 July 2001 Abstract Let G be a graph of order n. A graph G is called pancyclic if it contains a cycle of length k for every 3 6 k 6 n,and it is called vertex pancyclic if every vertex is contained in a cycle of length k for every 3 6 k 6 n. In this paper,we shall present di0erent su1cient conditions for graphs to be vertex pancyclic. ? 2002 Elsevier Science B.V. All rights reserved. 1. Terminology We consider ÿnite,undirected,and simple graphs G with the vertex set V (G) and the edge set E(G). The order of a graph G is denoted by n and the size by m, respectively. The complete graph of order n is denoted by Kn,and the complete bipartite graph with the partite sets A and B with |A| = p and |B| = q is denoted by Kp;q. A graph G is k-connected if G − S is connected for every subset S ⊆ V (G) with |S| 6 k − 1. A vertex v of a connected graph G is called a cut vertex if G − v is disconnected.
    [Show full text]
  • Graph Theory Graph Theory (II)
    J.A. Bondy U.S.R. Murty Graph Theory (II) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.
    [Show full text]
  • ON K-PATH PANCYCLIC GRAPHS
    Discussiones Mathematicae Graph Theory 35 (2015) 271–281 doi:10.7151/dmgt.1795 ON k-PATH PANCYCLIC GRAPHS Zhenming Bi and Ping Zhang Department of Mathematics Western Michigan University Kalamazoo, MI 49008, USA e-mail: [email protected] Abstract For integers k and n with 2 ≤ k ≤ n − 1, a graph G of order n is k-path pancyclic if every path P of order k in G lies on a cycle of every length from k +1 to n. Thus a 2-path pancyclic graph is edge-pancyclic. In this paper, we present sufficient conditions for graphs to be k-path pancyclic. For a graph G of order n ≥ 3, we establish sharp lower bounds in terms of n and k for (a) the minimum degree of G, (b) the minimum degree-sum of nonadjacent vertices of G and (c) the size of G such that G is k-path pancyclic Keywords: Hamiltonian, panconnected, pancyclic, path Hamiltonian, path pancyclic. 2010 Mathematics Subject Classification: 05C45, 05C38. 1. Introduction A Hamiltonian cycle in a graph G is a cycle containing every vertex of G and a graph having a Hamiltonian cycle is a Hamiltonian graph. The first theoretical result on Hamiltonian graphs occurred in 1952 and is due to Dirac [7]. Theorem 1.1 (Dirac). If G is a graph of order n ≥ 3 such that the minimum degree δ(G) ≥ n/2, then G is Hamiltonian. For a nontrivial graph G that is not complete, define σ2(G) = min{deg u + deg v : uv∈ / E(G)} where deg w is the degree of a vertex w in G.
    [Show full text]
  • A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas
    A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas Compiled by Thomas Zaslavsky Department of Mathematical Sciences Binghamton University (SUNY) Binghamton, New York, U.S.A. 13902-6000 E-mail: [email protected] Submitted: March 19, 1998; Accepted: July 20, 1998. Eighth Edition 6 September 2012 Mathematics Subject Classifications (2000): Primary 05-00, 05-02, 05C22; Secondary 05B20, 05B35, 05C07, 05C10, 05C15, 05C17, 05C20, 05C25, 05C30, 05C35, 05C38, 05C40, 05C45, 05C50, 05C60, 05C62, 05C65, 05C70, 05C75, 05C80, 05C83, 05C85, 05C90, 05C99, 05E25, 05E30, 06A07, 15A06, 15A15, 15A39, 15A99, 20B25, 20F55, 34C99, 51D20, 51D35, 51E20, 51M09, 52B12, 52C07, 52C35, 57M27, 68Q15, 68Q25, 68R10, 82B20, 82D30, 90B10, 90C08, 90C27, 90C35, 90C57, 90C60, 91B14, 91C20, 91D30, 91E10, 92D40, 92E10, 94B75. Colleagues: HELP! If you have any suggestions whatever for items to include in this bibliography, or for other changes, please let me hear from you. Thank you. Copyright c 1996, 1998, 1999, 2003{2005, 2009{2012 Thomas Zaslavsky the electronic journal of combinatorics #DS8 ii Index A 1 H 135 O 227 V 301 B 22 I 160 P 230 W 306 C 54 J 163 Q 241 X 317 D 77 K 171 R 242 Y 317 E 91 L 188 S 255 Z 321 F 101 M 201 T 288 G 113 N 221 U 301 [I]t should be borne in mind that incompleteness is a necessary concomitant of every collection of whatever kind. Much less can completeness be expected in a first collection, made by a single individual, in his leisure hours, and in a field which is already boundless and is yet expanding day by day.
    [Show full text]
  • Discrete Mathematics Forbidden Subgraphs for Chorded Pancyclicity
    Discrete Mathematics 340 (2017) 2878–2888 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Forbidden subgraphs for chorded pancyclicity Megan Cream a, Ronald J. Gould b, Victor Larsen c,* a Department of Mathematics, Spelman College, 350 Spelman Lane SW, Atlanta, GA 30314, United States b Department of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA 30322, United States c Department of Mathematics, Kennesaw State University, 1100 S. Marietta Parkway, Marietta, GA 30060, United States article info a b s t r a c t Article history: We call a graph G pancyclic if it contains at least one cycle of every possible length m, for Received 26 October 2016 3 ≤ m ≤ jV (G)j. In this paper, we define a new property called chorded pancyclicity. We Received in revised form 26 July 2017 explore forbidden subgraphs in claw-free graphs sufficient to imply that the graph contains Accepted 30 July 2017 at least one chorded cycle of every possible length 4; 5;:::; jV (G)j. In particular, certain Available online 1 September 2017 paths and triangles with pendant paths are forbidden. ' 2017 Elsevier B.V. All rights reserved. Keywords: Pancyclic Chorded cycle Forbidden subgraph Hamiltonian 1. Introduction In the past, forbidden subgraphs for Hamiltonian properties in graphs have been widely studied (for an overview, see [1]). A graph containing a cycle of every possible length from three to the order of the graph is called pancyclic. The property of pancyclicity is well-studied. In this paper, we define the notion of chorded pancyclicity, and study forbidden subgraph results for chorded pancyclicity.
    [Show full text]
  • Results on Pancyclc Graphs and Complete Hamiltonian Graphs
    International Journal of Engineering Applied Sciences and Technology, 2017 Vol. 2, Issue 5, ISSN No. 2455-2143, Pages 33-38 Published Online March-April 2017 in IJEAST (http://www.ijeast.com) RESULTS ON PANCYCLC GRAPHS AND COMPLETE HAMILTONIAN GRAPHS S.VENU MADHAVA SARMA T.V.PRADEEP KUMAR Research Scholar, Assistant Professor of Mathematics Department of Mathematics, ANU College of Engineering and Technology, Rayalaseema University, Acharya Nagarjuna University Kurnool, Andhra Pradesh. Guntur INDIA Abstract - In 1856, Hamiltonian introduced the In 1913, H. Dudeney mentioned a puzzle problem. Hamiltonian Graph where a Graph which is Eventhough the four color problem was invented it covered all the vertices without repetition and end was solved only after a century by Kenneth Appel with starting vertex. In this paper I would like to and Wolfgang Haken. prove that Relation between Complete Hamiltonian Graph and Pan-Cyclic Graphs This time is considered as the birth of Graph Theory. Key Words: Graph, Complete Graph, Bipartite Cayley studied particular analytical forms from Graph Hamiltonian Graph and Pan-Cyclic Graphs. differential calculus to study the trees. This had many implications in theoretical chemistry. This lead to the I. INTRODUCTION invention of enumerative graph theory. The origin of graph theory started with the problem Any how the term “Graph” was introduced by of Koinsber bridge, in 1735. Sylvester in 1878 where he drew an analogy between “Quantic invariants” and covariants of algebra and This problem lead to the concept of Eulerian Graph. molecular diagrams. Euler studied the problem of Koinsberg bridge and constructed a structure to solve the problem called In 1941, Ramsey worked on colorations which lead Eulerian graph.
    [Show full text]
  • Graph Theory Graph Theory (III)
    J.A. Bondy U.S.R. Murty Graph Theory (III) ABC J.A. Bondy, PhD U.S.R. Murty, PhD Universite´ Claude-Bernard Lyon 1 Mathematics Faculty Domaine de Gerland University of Waterloo 50 Avenue Tony Garnier 200 University Avenue West 69366 Lyon Cedex 07 Waterloo, Ontario, Canada France N2L 3G1 Editorial Board S. Axler K.A. Ribet Mathematics Department Mathematics Department San Francisco State University University of California, Berkeley San Francisco, CA 94132 Berkeley, CA 94720-3840 USA USA Graduate Texts in Mathematics series ISSN: 0072-5285 ISBN: 978-1-84628-969-9 e-ISBN: 978-1-84628-970-5 DOI: 10.1007/978-1-84628-970-5 Library of Congress Control Number: 2007940370 Mathematics Subject Classification (2000): 05C; 68R10 °c J.A. Bondy & U.S.R. Murty 2008 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or trans- mitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered name, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.
    [Show full text]
  • Graphs Which Have Pancyclic Complements
    I ntna. J. Math. Math. Si. Vol. (1978)177-185 177 GRAPHS WHICH HAVE PANCYCLIC COMPLEMENTS H. JOSEPH STRAIGHT Department of Mathematics SUNY College at Fredonla Fredonla, New York 14063 (Received January 23, 1978 and in revised form March 1978) ABSTRACT. Let p and q denote the number of vertices and edges of a graph G, respectively. Let A(G) denote the maximum degree of G, and G the complement of G. A graph G of order p is said to be pancycllc if G contains a cycle of each length n, 3 < n < p. For a nonnegative integer k, a connected graph G is said to be of rank k if q p i + k. (For k equal to 0 and i these graphs are called trees and unlcycllc graphs, respectively.) In 1975, I posed the following problem: Given k, find the smallest positive integer Pk' if it exists, such that whenever G is a rank k graph of order p <_ Pk and A(G) < p 2 then G is pancyclic. In this paper it is shown that a result by Schmeichel and Hakiml (2) guarantees that Pk exists. It is further shown that for k 0, i, and 2, Pk" 5, 6, and 7, respectively. i. INTRODUCTION. Throughout this paper the terminology of Behzad and Chartrand (i) will be 178 H. J. STRAIGHT followed. In particular, p and q shall denote the number of vertices and edges of a graph G, respectively. We let A(G) denote the maximum degree of G and G denote the complement of G.
    [Show full text]
  • Pancyclic and Bipancyclic Graphs
    SPRINGER BRIEFS IN MATHEMATICS John C. George Abdollah Khodkar W.D. Wallis Pancyclic and Bipancyclic Graphs 123 SpringerBriefs in Mathematics Series Editors Nicola Bellomo Michele Benzi Palle Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel G. George Yin Ping Zhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030 John C. George • Abdollah Khodkar • W.D. Wallis Pancyclic and Bipancyclic Graphs 123 John C. George Abdollah Khodkar Department of Mathematics Department of Mathematics and Computer Science University of West Georgia Gordon State College Carrollton, GA, USA Barnesville, GA, USA W.D. Wallis Department of Mathematics Southern Illinois University Evansville, IN, USA ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-319-31950-6 ISBN 978-3-319-31951-3 (eBook) DOI 10.1007/978-3-319-31951-3 Library of Congress Control Number: 2016935702 © The Author(s) 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.
    [Show full text]