# A simple criterion on degree sequences of graphs

Save this PDF as:

Size: px
Start display at page:

## Transcription

2 3514 A. Tripathi, H. Tyagi / Discrete Applied Mathematics 156 (2008) Main results We say that two sequences s 1 and s 2, of nonnegative integers, are graphically equivalent provided s 1 is graphic if and only if s 2 is graphic. Theorem 1 provides an instance of graphically equivalent sequences, much like Theorems H H and KW. However, in this case, for any d 0, we replace the largest term in the sequence by + d and also adjoin d 1 s. Theorem 1. Let {a 1, a 2, a 3,..., a n } be a sequence of positive integers such that n 1 a 1 a 2 a n, and let N a Let 1 r denote r occurrences of 1. Then the sequence {a 1, a 2, a 3,..., a n } is graphic if and only if the sequence {N, a 2, a 3,..., a n, 1 N a1 } is graphic. Proof. Suppose s := {a 1, a 2, a 3,..., a n } is a graphic sequence. To a graph G with degree sequence s, add N a 1 vertices and join each of these to the vertex of degree a 1. This gives a graph with degree sequence {N, a 2, a 3,..., a n, 1 N a1 }. Conversely, suppose s := {N, a 2, a 3,..., a n, 1 N a1 } is a graphic sequence. Applying Theorem KW N a 1 times, with a i = 1 each time, results in the sequence s, which must be graphic. This completes the proof. Theorem 1 is similar in nature to Theorems H H and KW, but gives a graphically equivalent sequence with a larger number of terms. The following definitions are central to our results, and repeatedly used in what follows. Definition 1. For a given sequence {a 1, a 2,..., a n } of positive integers such that a 1 a 2 a n, we denote by λ the largest positive integer i for which a i i and by R i the number of occurrences of i in the sequence. Our next result is derived by successive applications of Theorems 1 and H H. Theorem 2. The sequence {a 1, a 2,..., a n }, with a 1 a 2 a n, is graphic if and only if for 1 s λ 1 f(s) := [(n 1) a i (s i)r i ] 0, and f(λ) is even. Proof. Starting with the given sequence s 0, we first apply Theorem 1 and then the Havel Hakimi condition to obtain a graphically equivalent sequence s 1 : s 0 := {a 1, a 2,..., a n }; {n 1, a 2,..., a n, 1 n 1 a1 }; s 1 := {a 2 1, a 3 1, a 4 1,..., a n 1, 1 n 1 a1 }. Observe that this procedure transforms a sequence s 0 to a graphically equivalent sequence s 1 with a smaller maximum term. Thus this procedure must eventually lead to a sequence of 1 s, after discarding all 0 s at each step. However, to apply Theorem 1, we must ensure that none of the terms in the sequence are 0 and also that the maximum term is at least one less than the number of positive terms. So we must keep track of the number of positive terms at each stage in the procedure. Therefore we may equivalently assume that the sequence s 1 contains only positive terms. To do this, we may remove the R 1 (in s 0 ) terms in s 1 that are 0 and assume that the transformed sequence has only positive terms. Let N i denote the number of positive terms in the sequence s i obtained from s 0 after i iterations; thus N 0 = n and N 1 = (n 1 R 1 ) + (n 1 a 1 ) = 2(n 1) a 1 R 1. If we apply Theorem 1 followed by the Havel Hakimi condition to the sequence s 1, finally removing the zero terms, we get s 1 := {a 2 1, a 3 1, a 4 1,..., a n 1, 1 n 1 a1 } {N 1 1, a 3 1,..., a n 1, 1 n 1 a1, 1 N1 a 2 }; s 2 := {a 3 2, a 4 2,..., a n 2, 1 N1 a 2 } with N 1 positive terms; with N 2 positive terms, where N 2 = (n 2 R 1 R 2 ) + (N 1 a 2 ) = 3n 4 (a 1 + a 2 ) (2R 1 + R 2 ). Continuing in this manner, after i steps, removing the terms that are 0 at each stage, we arrive at with s i := {a i+1 i, a i+2 i,..., a n i, 1 Ni 1 a i +(i 2)} (2) N i = (i + 1)n 2i (a 1 + a a i ) (R i + 2R i ir 1 ) (3) positive terms. In order to continue this procedure at each stage, it is not only necessary that the maximum term in s i be at most N i 1 (otherwise we cannot apply Theorem 1), but also sufficient (otherwise s i cannot be graphic). Thus we need or a i+1 i N i 1 = (i + 1)n 2i (a 1 + a a i ) (R i + 2R i ir 1 ) 1, (i + 1)(n 1) (a 1 + a a i+1 ) + (R i + 2R i ir 1 ). (4) (1)

3 A. Tripathi, H. Tyagi / Discrete Applied Mathematics 156 (2008) With i = λ, we arrive at s λ := {a λ+1 λ, a λ+2 λ,..., a n λ, 1 N λ 1 aλ+(λ 2)}, (5) where N λ 1 = λn 2(λ 1) (a 1 + a a λ 1 ) (R λ 1 + 2R λ (λ 1)R 1 ). The condition on λ implies a λ+1 λ a λ. Thus s λ = {1 r }, where r = N λ 1 a λ + (λ 2), is graphic if and only if r = λ(n 1) (a 1 + a a λ ) (R λ 1 + 2R λ (λ 1)R 1 ) = f(λ) 0 (mod 2). (6) This completes the proof of our theorem. A theorem of Erdős and Gallai [4] gives another well known characterization of graphic sequences. Unlike the Havel Hakimi condition, this characterization requires verification of n inequalities, where n denotes the number of terms in the sequences. For the sake of reference, we state this result below: Theorem EG. [Erdős and Gallai, [4]] Let {a 1, a 2,..., a n } be a sequence of positive integers such that a 1 a 2 a n. Then the sequence {a 1, a 2,..., a n } is graphic if and only if n a i is even and the inequalities a i s(s 1) + min{s, a i } hold for each s with 1 s n. (7) It is interesting to note that inequalities (7) need not be verified for all s, but only for those s with a s > a s+1 and a s s in the notation of Theorem EG; see [11]. Since Theorems 2 and EG both require checking inequalities as a characterization for graphic sequences, it stands to reason that there is some relation between the inequalities (1) and (7), where in (7) we check only those s for which both a s > a s+1 and a s s. Indeed, as we shall presently show, they are algebraically the same. Fix s with 1 s λ 1. Then inequality (1) is the same as a i [(n 1) (s i)r i ] = s(s 1) + s(n s) Now for s < λ, i s implies a i a s s. Therefore min{s, a i } = s(n s) We have thus shown f(s) := (s i)r i. s 1 (s a i ) = s(n s) (s j)r j = s(n s) a i <s [(n 1) a i (s i)r i ] = s(s 1) + j=1 min{s, a i } (s i)r i. a i (8) for 1 s λ 1. This proves the equivalence of (1) and (7). In particular, Eq. (8) shows that f(λ) is even if and only if n a i is even since min{λ, a i } = a i for λ + 1 i n. This proves the desired equivalence between Theorem 2 and the stronger version of Theorem EG. We summarize these comments in the following. Remark 1. Theorem 2 can be strengthened to verifying (1) only for those s with both a s > a s+1 and a s s. Remark 2. We note that the above discussion shows that Theorem 2 gives an alternate proof of Theorem EG. The original proof of Theorem EG was simplified by Choudum in [2]. Sierksma and Hoogeveen showed that Hässelbarth s criterion [6] is equivalent to Theorem EG in [10]. However, Theorem 2 shows that Theorem EG can be reformulated solely in terms of the frequency of the items in the list, and in this manner, proved in a simpler though indirect manner. 3. Applications In this section, we apply the results in the previous section, mainly Theorem 2, to obtain some results on degree sequences and degree sets. An easy consequence of Theorem 2 is the following well known result: Proposition 1. Let n, r be integers, with n 1 and 0 r n 1. Then the sequence {r n } is graphic if and only if nr is even. In other words, there exists a r-regular graph of order n if and only if nr is even. Proof. Using the notation in Theorem 2, λ = r and the only nonzero R i is R r = n. Therefore the inequality in Eq. (1) is met, and the sequence {r n } is graphic if and only if (n 1)r r 2 = nr r(r + 1) nr 0 (mod 2).

4 3516 A. Tripathi, H. Tyagi / Discrete Applied Mathematics 156 (2008) Following Behzad and Chartrand in [1], we say that a graph has a perfect degree sequence if the sequence consists of distinct integers. Therefore the only perfect degree sequence of a graph of order n must be {0, 1, 2,..., n 1}. Perfect degree sequences cannot exist since both d(x) = 0 and d(y) = n 1 cannot hold in any (simple) graph of order n. Moreover, we say that a degree sequence is quasi-perfect if it has exactly one repeated entry. Behzad and Chartrand also showed that there are exactly two quasi-perfect graphic sequences of order n for each n > 1, and that the corresponding graphs are complements of each other. We explicitly show that there are exactly two quasi-perfect graphic sequences of order n for each n > 1. We use the following lemma, whose proof is trivial and omitted: Lemma 1. Let {a 1, a 2,..., a n } be a sequence of nonnegative integers. Then the sequence {a 1, a 2,..., a n } is graphic if and only if the sequence {n 1 a 1, n 1 a 2,..., n 1 a n } is graphic. Theorem 3 (Behzad and Chartrand [1]). For each n > 1, there are exactly two quasi-perfect graphic sequences of order n: n 1, n 2,..., n/2, n/2,..., 2, 1; n 2, n 3,..., (n 1)/2, (n 1)/2,..., 1, 0. Proof. Let n > 1. Since a quasi-perfect degree sequence of order n must have exactly one of 0, n 1, the two possibilities are 0, 1,..., n 2 with some r appearing twice, and 1, 2,..., n 1, again with some r appearing twice. We prove this result only in the case when n = 2m is even, the other case being similar. We prove that there is exactly one value of r in each case. Consider the sequence {2m 1, 2m 2,..., r+1, r, r, r 1,..., 1}, where 1 r 2m 1. With the notation of Theorem 2, we may denote this sequence by {a 1, a 2,..., a 2m }. Recall that λ denotes the largest positive integer i such that a i i and R i the number of occurrences of i in the sequence. The requirement on parity of the sum yields r m (mod 2). Moreover a m+1 m a m implies λ = m. Observe that f(s) = [(n 1) a 1 (s 1)R 1 ] + + [(n 1) a s 1 R s 1 ] + [(n 1) a s ] is the sum of an s-term arithmetic progression with first term 1 s and common difference 2 if 1 s r, unless the repeated term r is greater than m. Thus, if 1 s r m, f(s) = 0 and f(r + 1) = 1 since R r = 2, while f(2m r + 1) = 1 if r > m. Therefore, the inequality in (1) fails if r m, and by Theorem 2, the given sequence is not graphic in these cases. On the other hand, if r = m, f(s) = 0 for 1 s m, and so the sequence is graphic only in this case. By Lemma 1, the sequence {2m 2, 2m 3,..., r + 1, r, r, r 1,..., 1, 0} is graphic if and only if {2m 1, 2m 2,..., 2m r, 2m 1 r, 2m 1 r,..., 2, 1} is graphic. Therefore there is exactly one value of r for which quasiperfect sequences with a 0 are graphic, with the repeated value r given by 2m 1 r = m, so that r = m 1. This completes the assertion. The degree set of a graph is the set of (distinct) degrees of its vertices. A natural question that arises in the context of degree sets is to determine the least order among graphs with a given degree set. This was answered by Kapoor, Polimeni and Wall in [8], and a short proof of this was given by Tripathi and Vijay in [12]. While both these proofs construct the required graph by inducting on the size of the set, the following result explicitly provides a sequence, which we prove to be graphic by using Theorem 2. Theorem 4 (Kapoor, Polimeni and Wall [8]). Let S be a set of positive integers with maximum element. Then there exists a ( + 1)-term graphic sequence with set of distinct terms S. Moreover, there cannot be such a graphic sequence, with degree set S, having fewer terms. Proof. Let S = {a 1, a 2,..., a n }, where a 1 > a 2 > > a n 1. We construct a graphic sequence of order a with degree set S. Consider the sequence (a 1 ) m1, (a 2 ) m2,..., (a n ) mn, with m 1 = a n, m i = a n+1 i a n+2 i for 2 i n, i r, and m r = a (n+1)/2 a (n+1)/ , where r = (n + 1)/2. We verify that this sequence is graphic by using the strong version of Theorem 2, stated at the end of Section 2. We show this only in the case n is odd, the case when n is even being similar. Let σ i = m 1 + m m i for 1 i n. We need to show that the sequence has a terms, that λ = σ r 1, that f(σ i ) 0 for 1 i r 1, and that f(λ) is even. We observe that there are a n + (a n+1 i a n+2 i ) + 1 = a i=2 terms in the sequence.

5 A. Tripathi, H. Tyagi / Discrete Applied Mathematics 156 (2008) A simple calculation shows that σ i equals a n+1 i for 1 i r 1 and a n+1 i + 1 for r i n. Since a σr 1 = a r = a σr and σ r 1 = a n+1 r = a r, a λ = λ = a λ+1 for λ = σ r 1. Let 1 j r 1. Then σ j f(σ j ) = [a 1 a i (σ j i)r i ] = = m i (a 1 a i ) (σ j a i )R ai m i (a 1 σ j ) = σ j (a 1 σ j ) 0. A calculation similar to the one above shows that f(λ) = f(σ r 1) = a r (a 1 a r ). This must be even since a 1 is odd. That there cannot be a graphic sequence with fewer than + 1 terms and with degree set S is obvious. This completes the assertion. Acknowledgement The authors are grateful to all three referees for their careful reading and detailed comments, for pointing out a flaw in an argument, for pointing out a reference that resulted in a much simpler proof of Theorem 1, and above all for all their suggestions to modify the original manuscript. (9) References [1] M. Behzad, G. Chartrand, No graph is perfect, Amer. Math. Monthly 74 (1967) [2] S.A. Choudum, A simple proof of the Erdős Gallai theorem on graphic sequences, Bull. Austral. Math. Soc. 33 (1986) [3] R.B. Eggleton, Graphic sequences and graphic polynomials: A report, Infinite and Finite Sets 1 (1973) [4] P. Erdős, T. Gallai, Graphs with prescribed degrees of vertices, Mat. Lapok 11 (1960) (in Hungarian). [5] S.L. Hakimi, On the realizability of a set of integers as degrees of the vertices of a graph, J. SIAM Appl. Math. 10 (1962) [6] W. Hässelbarth, Die Verzweigtheit von Graphen, Match. 16 (1984) [7] V. Havel, A remark on the existence of finite graphs, Časopis Pěst. Mat. 80 (1955) (in Czech). [8] S.F. Kapoor, A.D. Polimeni, C.E. Wall, Degree sets for graphs, Fund. Math. 95 (1977) [9] D.J. Kleitman, D.L. Wang, Algorithms for constructing graphs and digraphs with given valences and factors, Discrete Math. 6 (1973) [10] G. Sierksma, H. Hoogeveen, Seven criteria for integer sequences being graphic, J. Graph Theory 15 (1991) [11] A. Tripathi, S. Vijay, A Note on a Theorem of Erdős and Gallai, Discrete Math. 65 (2003) [12] A. Tripathi, S. Vijay, A short proof of a theorem on degree sets of graphs, Discrete Appl. Math. 155 (2007)

### Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z

FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER Z DANIEL BIRMAJER, JUAN B GIL, AND MICHAEL WEINER Abstract We consider polynomials with integer coefficients and discuss their factorization

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### On the Sum Necessary to Ensure that a Degree Sequence is Potentially H-Graphic

On the Sum Necessary to Ensure that a Degree Sequence is Potentially H-Graphic Michael J. Ferrara 1,2 Timothy D. LeSaulnier 3 Casey K. Moffatt 1 Paul S. Wenger 4 September 2, 2014 Abstract Given a graph

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Reading 7 : Program Correctness

CS/Math 240: Introduction to Discrete Mathematics Fall 2015 Instructors: Beck Hasti, Gautam Prakriya Reading 7 : Program Correctness 7.1 Program Correctness Showing that a program is correct means that

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

GENERATING SETS KEITH CONRAD 1 Introduction In R n, every vector can be written as a unique linear combination of the standard basis e 1,, e n A notion weaker than a basis is a spanning set: a set of vectors

### Mathematical Induction

Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,

### CONTRIBUTIONS TO ZERO SUM PROBLEMS

CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### CHAPTER 5: MODULAR ARITHMETIC

CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called

### Chapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1

Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS

POLYNOMIAL RINGS AND UNIQUE FACTORIZATION DOMAINS RUSS WOODROOFE 1. Unique Factorization Domains Throughout the following, we think of R as sitting inside R[x] as the constant polynomials (of degree 0).

### The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Subsets of Euclidean domains possessing a unique division algorithm

Subsets of Euclidean domains possessing a unique division algorithm Andrew D. Lewis 2009/03/16 Abstract Subsets of a Euclidean domain are characterised with the following objectives: (1) ensuring uniqueness

### (Refer Slide Time: 01.26)

Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture # 27 Pigeonhole Principle In the next few lectures

### Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties

Lecture 1 (Review of High School Math: Functions and Models) Introduction: Numbers and their properties Addition: (1) (Associative law) If a, b, and c are any numbers, then ( ) ( ) (2) (Existence of an

### Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### 8.7 Mathematical Induction

8.7. MATHEMATICAL INDUCTION 8-135 8.7 Mathematical Induction Objective Prove a statement by mathematical induction Many mathematical facts are established by first observing a pattern, then making a conjecture

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### 3.1 The Definition and Some Basic Properties. We identify the natural class of integral domains in which unique factorization of ideals is possible.

Chapter 3 Dedekind Domains 3.1 The Definition and Some Basic Properties We identify the natural class of integral domains in which unique factorization of ideals is possible. 3.1.1 Definition A Dedekind

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### Prime Numbers and Irreducible Polynomials

Prime Numbers and Irreducible Polynomials M. Ram Murty The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry.

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### The finite field with 2 elements The simplest finite field is

The finite field with 2 elements The simplest finite field is GF (2) = F 2 = {0, 1} = Z/2 It has addition and multiplication + and defined to be 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 0 0 = 0 0 1 = 0

### A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Euler Paths and Euler Circuits

Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

### Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2 Ameya Pitale, Ralf Schmidt 2 Abstract Let µ(n), n > 0, be the sequence of Hecke eigenvalues of a cuspidal Siegel eigenform F of degree

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

### An example of a computable

An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

### Kevin James. MTHSC 412 Section 2.4 Prime Factors and Greatest Comm

MTHSC 412 Section 2.4 Prime Factors and Greatest Common Divisor Greatest Common Divisor Definition Suppose that a, b Z. Then we say that d Z is a greatest common divisor (gcd) of a and b if the following

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### MOP 2007 Black Group Integer Polynomials Yufei Zhao. Integer Polynomials. June 29, 2007 Yufei Zhao yufeiz@mit.edu

Integer Polynomials June 9, 007 Yufei Zhao yufeiz@mit.edu We will use Z[x] to denote the ring of polynomials with integer coefficients. We begin by summarizing some of the common approaches used in dealing

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### Taylor Series and Asymptotic Expansions

Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious.

### Sign pattern matrices that admit M, N, P or inverse M matrices

Sign pattern matrices that admit M, N, P or inverse M matrices C Mendes Araújo, Juan R Torregrosa CMAT - Centro de Matemática / Dpto de Matemática Aplicada Universidade do Minho / Universidad Politécnica

### Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.

Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,

### Mathematical Induction

Chapter 2 Mathematical Induction 2.1 First Examples Suppose we want to find a simple formula for the sum of the first n odd numbers: 1 + 3 + 5 +... + (2n 1) = n (2k 1). How might we proceed? The most natural

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

### Metric Spaces. Chapter 1

Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

### Induction and Recursion

Induction and Recursion Winter 2010 Mathematical Induction Mathematical Induction Principle (of Mathematical Induction) Suppose you want to prove that a statement about an integer n is true for every positive

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1, No.3,August 2013

FACTORING CRYPTOSYSTEM MODULI WHEN THE CO-FACTORS DIFFERENCE IS BOUNDED Omar Akchiche 1 and Omar Khadir 2 1,2 Laboratory of Mathematics, Cryptography and Mechanics, Fstm, University of Hassan II Mohammedia-Casablanca,

### ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS

ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for

### Bipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES. 1. Introduction

F A S C I C U L I M A T H E M A T I C I Nr 51 2013 Bipan Hazarika ON ACCELERATION CONVERGENCE OF MULTIPLE SEQUENCES Abstract. In this article the notion of acceleration convergence of double sequences

### Mean Value Coordinates

Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### A simple Havel Hakimi type algorithm to realize graphical degree sequences of directed graphs

simple Havel Hakimi type algorithm to realize graphical degree sequences of directed graphs Péter L. Erdős and István Miklós. Rényi Institute of Mathematics, Hungarian cademy of Sciences, Budapest, PO

### AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC

AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following.

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

### About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### ON THE FIBONACCI NUMBERS

ON THE FIBONACCI NUMBERS Prepared by Kei Nakamura The Fibonacci numbers are terms of the sequence defined in a quite simple recursive fashion. However, despite its simplicity, they have some curious properties

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

Chapter 1 A Pri Characterization of T m e Pairs w in Proceedings NCUR V. (1991), Vol. I, pp. 362{366. Jerey F. Gold Department of Mathematics, Department of Physics University of Utah DonH.Tucker Department

### 5 The Beginning of Transcendental Numbers

5 The Beginning of Transcendental Numbers We have defined a transcendental number (see Definition 3 of the Introduction), but so far we have only established that certain numbers are irrational. We now

### On the Union of Arithmetic Progressions

On the Union of Arithmetic Progressions Shoni Gilboa Rom Pinchasi August, 04 Abstract We show that for any integer n and real ɛ > 0, the union of n arithmetic progressions with pairwise distinct differences,

### Math 231b Lecture 35. G. Quick

Math 231b Lecture 35 G. Quick 35. Lecture 35: Sphere bundles and the Adams conjecture 35.1. Sphere bundles. Let X be a connected finite cell complex. We saw that the J-homomorphism could be defined by

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### 12 Greatest Common Divisors. The Euclidean Algorithm

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 12 Greatest Common Divisors. The Euclidean Algorithm As mentioned at the end of the previous section, we would like to

### Product irregularity strength of certain graphs

Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

### Some Polynomial Theorems. John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.

Some Polynomial Theorems by John Kennedy Mathematics Department Santa Monica College 1900 Pico Blvd. Santa Monica, CA 90405 rkennedy@ix.netcom.com This paper contains a collection of 31 theorems, lemmas,

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Invertible elements in associates and semigroups. 1

Quasigroups and Related Systems 5 (1998), 53 68 Invertible elements in associates and semigroups. 1 Fedir Sokhatsky Abstract Some invertibility criteria of an element in associates, in particular in n-ary

### ELEMENTARY NUMBER THEORY AND METHODS OF PROOF

CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF SECTION 4.4 Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem Copyright Cengage Learning. All rights reserved.

### THE 2-ADIC, BINARY AND DECIMAL PERIODS OF 1/3 k APPROACH FULL COMPLEXITY FOR INCREASING k

#A28 INTEGERS 12 (2012) THE 2-ADIC BINARY AND DECIMAL PERIODS OF 1/ k APPROACH FULL COMPLEXITY FOR INCREASING k Josefina López Villa Aecia Sud Cinti Chuquisaca Bolivia josefinapedro@hotmailcom Peter Stoll

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### Linear Span and Bases

MAT067 University of California, Davis Winter 2007 Linear Span and Bases Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (January 23, 2007) Intuition probably tells you that the plane R 2 is of dimension

### DETERMINANTS. b 2. x 2

DETERMINANTS 1 Systems of two equations in two unknowns A system of two equations in two unknowns has the form a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2 This can be written more concisely in

### RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES

RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES JONATHAN CHAPPELON, LUIS PEDRO MONTEJANO, AND JORGE LUIS RAMÍREZ ALFONSÍN Abstract Let K [k,t] be the complete graph on k vertices from which a set of edges,

### Regular Languages and Finite Automata

Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a