(a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict.)

Transcription

1 1 Enumeration 11 Basic counting principles 1 June 2008, Question 1: (a) How many pairs (A, B) are there with A, B [n] and A B? (The inclusion is required to be strict) n/2 ( ) n (b) Find a closed form for 2k 2 June 2008, Question 2: k=0 (a) Let d n be the number of derangements of [n] Prove that d n is odd if and only if n is even (By convention we define d 0 =1) (b) Suppose that A = (a ij ) is an n n matrix with zeroes on the main diagonal such that a ij = ±1 for all i j Prove that if n is even then det(a) 0 3 January 2005, Question 1:Let d n be the number of derangements of [n] = {1, 2,, n} that is permutations π of [n] such that π(i) i for all i [n] (a) Prove that d n satisfies the recurrence relation d n = (n 1)(d n 1 + d n 2 ) for all n 2 (b) By rewriting this recurrence as d n nd n 1 = (d n 1 (n 1)d n 2 ), or otherwise, prove that d n = nd n 1 + ( 1) n (c) Prove that d n is even if and only if n is odd 4 January 2008, Question 1: (a) How many increasing functions are there which map [n] = {1, 2,, n} to [m]? [We do not require that the functions are strictly increasing] m ( ) n k (b) Give a closed form expression for m k 5 January 2007, Question 1: k=0 (a) Give a combinatorial proof of the following identity: ( ) n = k n 1 i=k 1 ( ) i k 1 [Proofs by other techniques will receive little credit] 1

2 (b) Find a closed form for the expression 6 June 2007, Question 1: (a) Find a closed form for the sum n i=0 ( i 2 n i=1 )( n i ( ) i n n i i (b) How many subsets of {1, 2,, n} of size k contain no pair of elements i, j with i j 3? n ( )( ) k n 7 June 2006, Question 1(a): Simplify m k 8 January 2006, Question 3a: Compute k=m n/2 k=0 ( ) n 2k 9 June 2004, Question 4: Give combinatorial proofs of the following facts [Other proof techniques will receive little credit] (a) s(n + 1, k) = ns(n, k) s(n, k 1) where s(n, k) is the Stirling number of the first kind, so ( 1) n k s(n, k) is the number of permutations of {1, 2,, n} with k cycles n ( ) n (b) k = n2 n 1 k k=1 (c) d n = (n 1)(d n 1 +d n 2 ), where d n is the number of derangements of {1, 2, n} 10 June 2005, Question 6: Prove the following (a) Fn 2 F n+1 F n 1 = ( 1) n where the F n are the Fibonacci numbers, the solution the recurrence F n = F n 1 + F n 2, F 0 = 1, F 1 = 1 n ( ) n (b) i = n2 n 1 i i=0 (c) The number of solutions in integers to x 1 + x x k = n, x i 1 is ( n 1 k 1) 11 January 2003, Question 1: Give combinatorial proofs of the following identities (Other styles of proof will receive little credit) (a) (b) k ( ) ( ) n + i n + k + 1 = i k n ( ) n i = n2 n 1 i i=0 i=0 2 )

3 12 January 2003, Question 2b: In order to become a bridge life master (in the American Contract Bridge League) you need a total of 300 points, of which at least 50 must be black, at least 50 silver, and at least 25 must be gold The only other kind of point available is platinum Assume that one can only win whole numbers of each kind of point, how many distributions of 300 points are possible which qualify one as a life master? 13 January 2002, Question 7: Give combinatorial proofs that (a) D n = (n 1)D n 1 + (n 1)D n 2 and (b) S(n, k) = ks(n 1, k) + S(n 1, k 1) where D n is the number of derangements of {1, 2, n} and S(n, k) is the number of partitions of {1, 2,, n} into k non-empty subsets 14 January 2002, Question 8: Determine the number of ways selecting r distinct integers out of the first n positive integers such that the selection does not include 2 consecutive integers 15 June 2002, Question 2: How many solutions in integers are there to the system { x 1, x 2,, x k 1 x 1 + x x k = m? Prove your answer How many subsets of {1, 2, n} contain no subset of the form {i, i + 1}, 1 i n 1? 3

4 12 Recurrence relations 1 January 2006, Question 3b: Let c n be the number of monotonically increasing functions from {1, 2,, n} to itself with the property that f(i) i for all i Find a recurrence relation satisfied by the sequence (c n ) 1 2 June 2005, Question 7a: Find the general form of the solution to the recurrence relation a n = a n 1 + 6a n January 2005, Question 3: Solve the recurrence relation a n = 4a n 1 a n 2 + 6a n 3 a 0 = 2, a 1 = 2, a 2 = 10 4 June 2004, Question 3: Solve the recurrence relation 5 June 2003, Question 3: a n = 3a n 2 2a n 3 a 0 = 3, a 1 = 1, a 2 = 8 (a) Find the general solution the the following recurrence a n + 2a n 1 15a n 2 = 3 n (b) A permutation π of [n] is called connected if there does not exist any i < n for which π maps [i] to [i] Let c n be the number of connected permutations of [n] Prove that n c i (n i)! = n! i=1 6 January 2003, Question 2(a): Find the general solution of the following recurrence a n + 5a n 1 14a n 2 = 3 n 7 January 2002, Question 5: Solve the following recurrence relation: f(n) = 2f(n 1) + f(n 2) 2f(n 3) for n 3 where f(0) = 1, f(1) = 2, f(2) = 0 8 June 2002, Question 1: What is the general solution to the recurrence x n + 3x n 1 4x n 3 = 9? 4

5 13 Ordinary and Exponential Generating Functions, Exponential Formula 1 Use ordinary generating functions to evaluate the sum r ( )( ) n n ( 1) k k r k k=0 2 Use ordinary generating functions to evaluate the sum n ( ) 2 n k k 3 Use exponential generating functions to evaluate the sum n ( ) n ( 1) k 1 k 2 n k k k=0 k=0 4 Using exponential generating functions, determine the number of n digit numbers with all digits odd such that 1 and 3 each occur a nonzero even number of times 5 Let g n denote the number of ways in which n students can partition themselves into clubs that each have a president and vice president (the president and vice president must be different students; no club can be formed with fewer than two students) Determine the exponential generating function for g 14 Principle of Inclusion-Exclusion ***Almost all of the following questions have the following part (a): State and prove the Principle of Inclusion-Exclusion*** 1 January 2007, Question 2: In a small town n married couples attend a town meeting, and each of these 2n people want to speak exactly once In how many ways can the speakers be scheduled if we insist no married couple speaks in consecutive slots? 2 June 2007, Question 2: A class of n students walk to the park one day in single file On the way back the same students want to walk in single file again, but they want to walk in an order such that no one sees the same person in front of them as they did on the way there In how many ways can the students be lined up satisfying this constraint? 3 June 2006, Question 1: (a) Simplify n k=m ( k m)( n k) (b) Give, with proof, an expression for the number of surjective functions from a set of size n to one of size k 5

6 4 January 2006, Question 1: How many surjective maps are there from {1, 2,, n} to {1, 2, k}? Justify your answer 5 June 2005, Question 10: Some married couples arrive at a dinner party How many different ways are there to seat them around a circular table such that no husband and wife sit next to each other? 6 January 2005, Question 5: You are to make a necklace from n different pairs of beads The beads in a pair have the same colour but different shapes In how many ways can you make the necklace so that no two beads of the same colour are adjacent? 7 June 2004, Question 1: Prove that the number of partitions of an n-element set into k nonempty parts satisfies S(n, k) = 1 k! k ( ) k ( 1) i (k i) n i i=0 8 June 2003, Question 3: A derangement is a permutation π with the property that π(i) i Prove that the number of derangements of [n] is the nearest integer to n!/e 9 January 2003, Question 4: Give, with justification a formula for the number of surjections (onto functions) from [n] = {1, 2,, n} to [k] = {1, 2,, k} 10 The chromatic function of a graph G is the function χ G : N N given by: χ G (k) = {c : V (G) {1, 2, k} : c is a proper vertex colouring} Prove using the Principle of Inclusion - Exclusion that the chromatic function is in fact a polynomial in k Show moreover, that if G has n vertices and m edges then the leading terms of χ G are χ G (k) = k n mk n Burnside s Lemma ***All of the following questions have the following part (a): State and prove Burnside s Lemma concerning the number of orbits of a group action [You may assume without proof that if a group G acts on a set X then Stab(x) Orb(x) = G, where Stab(x) is the stabilizer of x X and Orb(x) is its orbit]*** 1 June 2008, Question 3: How many different ways are there to color the edges of K 4 with two colors, red and blue? Two colorings are the same if some permutation of the vertices takes one to the other [One can think of this problem as counting the number of isomorphism classes of graphs on four vertices] 2 January 2008, Question 4: Some identity cards are to be made taking square cards ruled into a 7 7 grid and punching out two of the squares The cards can be inserted into a scanner with any orientation How many different identity cards can be produced in this way? 6

7 3 January 2007, Question 5: How many ways are there to make a 9 bead necklace out of red, white, and black beads if two necklaces which are rotations of each other are considered to be the same, but necklaces which are reflections of one another are considered distinct? 4 June 2007, Question 5: How many essentially different ways are there to color the edges of a regular octohedron with two colors? [Octahedral dice are available on request from the proctor] 5 June 2006, Question 3: Compute the number of distinguishable ways there are to color the vertices of a solid triangular prism using 3 colors 6 January 2006, Question 4: How many distinguishable ways are there to color the faces of an octahedral die with the colors red, white, and blue? [Octohedral dice are available on request from the proctor] 7 June 2005, Question 9: How many essentially different ways are there to the paint the surface of a cube with three different colours of paint? 8 January 2005, Question 2: Some identity cards are to be made by taking square cards ruled into a 5 5 grid and punching out two of the squares How many different cards can be produced this way? 9 June 2004, Question 2: How many ways are there to colour the squares in a Tic-Tac- Toe grid with the colours red, white, and blue, if two colourings which differ by a rotation or a reflection are considered the same? 10 June 2003, Question 5: How many different necklaces can e made from n beads each of which is black or white? Two necklaces are considered the same if they only differ by a rotation, but not if they differ by a reflection 11 January 2003, Question 10: How many different ways are there to colour the small triangles in the figure below with the colours red, white and blue, if rotations and reflections count as the same colouring? 12 June 2002, Question 3: How many essentially different ways are there to paint the corners of a solid cube with 5 colours? 7

8 16 The basic probabilistic method and linearity of expectation 1 (a) Use a random partition of the vertices to prove that every graph has a bipartite subgraph with at least half of its edges (b) Prove that every graph G with m edges that has a matching with k edges has a bipartite subgraph with at least (m + k)/2 edges 2 An instance of SATISFIABILITY is a list of clauses where each clause is a set of literals (a literal is a variable or its negation) A satisfying assignment sets each variable TRUE or FALSE so that each clause has at least one true literal Given that all clauses have size k, prove that the minimum number of clauses in an unsatisfiable instance is 2 k 3 A tournament is an orientation of K n A subset of vertices in a tournament is beaten if some vertex beats every element of it Prove that if ( n k) (1 2 k ) n k < 1 then there exists a tournament on n vertices such that every k-set is beaten 4 If G is a graph then α(g) v V 1 d(v) + 1 (Hint: Pick a random ordering on the vertices) 8

9 2 Discrete Structures, Coding Theory, Design Theory, Posets 21 Coding Theory Basics 1 January 2006, Question 2: Let C be a binary code of length n with minimum distance d > n/2 having M codewords Enumerate C as C = {c 1, c 2,, c M } (a) Prove that M(M 1)d M i=1 M j=1 d(c i, c j ) nm 2 2 (b) Deduce that C has at most 2d/(2d n) codewords (This bound is called the Plotkin bound) 22 Linear Codes 1 January 2008, Question 5: Given integers 0 < d n and a prime power q prove that there exists a linear code C F n q of minimum distance at least d, containing at least / ( d 1 ( ) n q )(q n 1) i i codewords 2 January 2007, Question 4: i=0 (a) Let C be a binary linear code and let E be the subset of C consisting of those codewords having even length Prove that E is either C or C /2 (b) Prove that given positive integers n, q, d there is a q-ary code of lenght N and minimum distance d having at least d 1 i=0 ( n i q n ) (q 1) i codewords [Note: we are not requiring this code to be linear] 23 Sphere packing 1 June 2004, Question 5: (a) State and prove the Hamming (or sphere packing ) bound on the number of words in a q-ary code with length n and minimum distance at least d (b) Prove that for binary codes the Hamming bound is always at least as strong as the Singleton bound 24 Generalized Reed Solomon Codes 9

10 25 Posets 1 January 2008, Question 3: Suppose 0 < t < n/2 and that A P(n) is an antichain with A t for all A A Define { ( ) } [n] A t = B : there is some A A with A B t Prove that A A t 2 January 2007, Question 3: (a) Prove that if P is a poset such that no chain has length more than k then P can be written as the union of at most k antichains [Hint: This is not Dilworth s theorem] (b) Prove that a sequence of distinct real numbers of length n 2 must contain either an increasing subsequence of length n + 1 or a decreasing subsequence of length n June 2007, Question 3: State and prove Sperner s lemma concerning the largest size of an antichain in the power set of {1, 2,, n} 4 June 2006, Question 4: Let c 1, c 2,, c n 0 be real numbers such that n i=1 c i = 1 Prove that the collection A = {A [n] : i A c i > 1 2 } has size at most 2n 1 5 January 2006, Question 2: State and prove the Erdos-Ko-Rado Theorem concerning the maximum size of an intersecting family of k-sets 6 January 2002, Question 6: A family F of subsets of X is intersecting if A, B F A B (a) Prove that an intersecting family F of subsets of X = {1, 2, n} satisfies F 2 n 1 (b) Prove that any intersecting family F of X = {1, 2,, n} can be extended to an intersecting family of size 2 n 1 7 Prove that a poset of size greater than mn has a chain of size greater than m or an antichain of size greater than n Use this to prove the Erdos-Szekeres Theorem: every list of mn + 1 distinct integers has an increasing sublist with more than m elements or a decreasing sublist with more than n elements 8 A family of sets is union free if it does not have two distinct members whose union is a third member of the family Moser asked for the maximum size of a union-free subfamily that can be guaranteed to exist in any family of n sets Use Dilworth s Theorem to give a short proof that every family of n distinct sets contains a union-free family of size at least n 9 Prove that the largest antichain of subsets of [n] consisting of pairs of complementary sets has size 2 ( n 1 n/2 ) (Hint: Use the Erdos-Ko-Rado Theorem) 10

11 10 Prove the following dual of Dilworth s Theorem: If P is a finite poset, then the maximum size of a chain in P equals the minimum number of antichains needed to cover the elements of P (Hint: The set of maximal elements is an antichain Remember that there are two inequalities needed to be shown) 11 Let n 2k and let A 1,, A m be a family of k-element subsets of [n] such that A i A j [n] for all i, j Show that m (1 k )( n n k) (Hint: Apply the Erdos-Ko-Rado theorem to the complements A i ) 12 Let F be a k-uniform family, and suppose that it is intersection free, ie, that A B C for any three sets A, B, and C of F Prove that F 1 + ( k k/2 ) (Hint: Fix a set B 0 F and observe that {A B 0 : A F, A B 0 } is an antichain over B 0 ) 13 Let x 1,, x n be real numbers, x i 1 for each i and let S be the set of all numbers which can be obtained as linear combinations α 1 x α n x n with α i { 1, +1} Let I = [a, b) be any interval (in the real line) of length b a = 2 Show that I S ( n n/2 ) 26 Designs 1 June 2003, Question 2 & January 2003, Question 3 & January 1995, Question 9: Let B = {B 1, B 2,, B b } be a family of subsets (called blocks) of X = [v] Further, assume that each unordered pair {j, m} X occurs in exactly λ > 0 blocks Prove that if each i X belongs to strictly more than λ of the blocks then b v 2 January 2002, Question 9: Let B = {B 1, B 2,, B b } be a family of subsets (called blocks) of a v-set X = {x 1, x 2,, x v } Further, assume that each unordered pair {x j, x m }, 1 j < m v, occurs in exactly λ > 0 blocks Prove that if λ < B i < v, for 1 i v, then b v (Hint: Use an incidence matrix and determinant) 3 In a block design, block sizes can vary Construct blocks (subsets of [v] for some v) such that any two elements of [v] appear together in one block, all elements appear equally often, and the block sizes are not all equal 4 A Steiner quadruple system (SQS) is a collection Q of 4-element subsets of X such that for any 3-subset T of X there exists a unique Q Q such that T Q The integer n = X is called the order of Q (a) Prove that if Q is an SQS of order n, then Q = n(n 1)(n 2)/24 (b) Let V be a (finite dimensional) vector space over F 2 Define { ( ) V Q = Q : } v = 0 4 v Q Prove that Q is an SQS on the ground set V 11

12 5 When q 2 a family of (q + 1)-sets is the family of lines of a projective plane of order q if and only if the family is a symmetric (q 2 + q + 1, q + 1, 1) design 6 Let B be a balanced incomplete block design with parameters for v, k, λ, b, r whose set of varieties is V = {x 1,, x v } and whose blocks are B{B 1,, B b } For each block B i, let B i = V \ B i Let B c denote the collection of subsets B 1,, B b of V Prove that B c is a block design if b 2r + λ > 0, and determine its parameters v, k, λ, b, r 12

13 3 Graph Theory 31 Basic Concepts of Graphs 1 January 2007, Question 6: Recall that a trail is a walk in a graph that does not use any edge more than once A graph is called randomly Eulerian from vertex x if every maximal trail starting at x is an Euler circuit Prove that G is randomly Eulerian starting at x if and only if G has an Euler circuit and x is contained in every cycle of G 2 June 2006, Question 6: Prove that every non-trivial tree contains at least two maximal independent sets, with equality only for stars 3 January 2006, Question 1: (a) Prove that a connected graph on n vertices has n edges if and only if it contains exactly one cycle (b) Let n 3 and suppose that G is an n vertex graph with the property that for all v V (G) the graph G \ {v} is a tree Determine the number of edges in G, and thereby determine G 4 June 2005, Question 1: [Note that in both parts we are counting different graphs, not non-isomorphic graphs] (a) Prove that the number of graphs with vertex set {1, 2,, n} is 2 (n 2) (b) Prove that the number of graphs with vertex set {1, 2,, n} such that all the vertex degrees are even is 2 (n 1 2 ) 5 January 2005, Question 7: Let d = d(g) = 2e(G)/n(G) > 0 be the average degree of a graph G (a) Prove that G has a subgraph H with δ(h) > d/2 [Hint: consider successively deleting vertices of degree at most d/2] (b) Prove that for all c < 1/2 there exists a graph G with no subgraph H having [Hint: Consider K 1,n ] δ(h) > cd(g) 6 June 2004, Question 8: Prove that every graph has a bipartition V (G) = X Y with the property that e(x, Y ) e(g)/2 Further, show that if G is 3-regular then we can achieve e(x, Y ) n(g) = 2e(G)/3 7 June 2003, Question 6: (a) Suppose that G is a k-regular graph with k 1, having n vertices Prove that α(g) n/2 13

14 (b) Suppose that T is a tree having n vertices Prove that α(t ) n/2 with equality if and only if T has a perfect matching 8 January 2003, Question 6: Let T be a tree We say that T splits oddly if for every edge e E(T ) the two components of T e each have an odd number of vertices Prove that the vertices of T all have odd degree if and only if T splits oddly 9 January 2002, Question 3: (a) State and prove a necessary and sufficient condition for a connected graph G to be Eulerian in terms of its degrees (b) By finding an Euler circuit in a suitably defined directed graph, construct a circular binary sequence S such that each binary word of length 4 appears exactly once as one moves along the sequence S 10 June 2002, Question 9: Let G be a graph with vertex set V = {v 1, v 2,, v n } The adjacency matrix of G is the n n matrix A = [a ij ] with { 1 v i is adjacent to v j in G a ij = 0 otherwise Prove that the ij th entry of A k is the number of walks of length k in G from v i to v j 11 June 2002, Question 10: Let G be a connected bipartite graph which is k-regular for some k 2 Prove that G is bridgeless 12 June 2007, Question 6: Consider a family A = {A 1, A 2,, A n } of distinct subsets of some n-set X Define a graph G on A with an edge between A i and A j if A i A j = 1 If A i A j = {x} then we label the edge A i A j with x Prove that there is a forest F G whose edges include all the labels used on edges of G Deduce that there exists some x X for which the sets A 1 \ {x}, A 2 \ {x},, A n \ {x} are distinct 13 June 2007, Question 10: In this question we consider directed graphs (digraphs) with no loops and no multiple edges (but we do allow both x y and y x) A monotone tournament is an orientation of a complete graph in which (for some ordering of the vertices) the ordering on the edges is from the smaller to the larger vertex A complete digraph is a digraph in which both x y and y x are edges for every x, y in its vertex set Given m 1 prove that if N is sufficiently large then every digraph on N vertices contains a subset of size m which induces either an empty digraph, a complete digraph, or a monotone tournament 32 Matchings 1 June 2008, Question 6: Prove the Konig-Egerváry Theorem from the vertex version of Menger s Theorem 14

15 2 June 2008, Question 7(a): Prove that every tree has at most one perfect matching 3 June 2007, Question 9: Prove that a tree T has a perfect matching if and only if the number of odd components in G v is 1 for every vertex v of G 4 June 2006, Question 2: Let G be a bipartite graph with bipartition A, B that contains a matching from A to B (a) Prove that for some vertex a A it is the case that for all edges ab E(G) there is a matching from A to B that contains ab [Hint: It may be helpful to consider whether or not there is subset C A having exactly C neighbours] (b) Deduce that if all vertices in A have degree d then the number of matchings from A to B in G is at least d! if d A and at least d(d 1)(d 2) (d A + 1) if d > A 5 January 2006, Question 4: Suppose that X is a set of size mn and (A i ) m 1, (B i ) m 1 are partitions of X into m sets of size n Prove that one can renumber the sets B i in such a way that A i B i for i = 1, 2, m 6 January 2006, Question 5: (a) State Tutte s Theorem concerning graphs having a 1-factor (b) Prove that every connected, bridgeless, 3-regular graph has a 1-factor (c) Find a connected 3-regular graph with no 1-factor 7 June 2004, Question 6: Let A 1, A 2,, A n be finite sets and d 1, d 2,, d n non-negative integers Prove that there are disjoint subsets D i A i with D i = d i if and only if for all I {1, 2, n} we have A i d i i I i I 8 June 2003, Question 4: (a) State Hall s theorem concerning systems of distinct representatives (b) A positional game consists of a set X of positions and a set W P(X) of winning sets of positions [For instance in Tic-Tac-Toe the positions are X = {1, 2, 3} 2 and the wining sets are those which contain a line] Two players alternately select positions from X until one player s set of selected positions is in W (No position can be selected twice) Suppose that W a for all W W and no x X belongs to more than b winning sets Prove that the second player can force a draw if a 2b [Hint: consider the bipartite graph with vertex classes X and two disjoint copies of W Join x X to W W whenever x W] 9 January 2003, Question 5: (a) State Hall s theorem concerning systems of distinct representatives 15

16 (b) A permutation matrix is {0, 1}-matrix having exactly one 1 in every row and column Prove that an n n matrix A with non-negative integer entries is a sum of permutation matrices if and only if all the row and column sums are equal 10 June 2002, Question 6: Suppose that G is a bipartite graph with bipartition (X, Y ) Define the deficiency of a subset S X to be def(s) = S N(S) Prove that the maximum size of a matching in G is 33 Connectivity X max{def(s) : S X} 1 June 2007, Question 7: Prove that tree T having 2k endvertices contains k edge-disjoint paths joining all the endvertices in pairs Deduce if x is a vertex in a graph G with degree 2k and x is not a cutvertex of G then x is contained in k edge-disjoint cycles 2 June 2006, Question 7: (a) State the Max Flow/Min Cut theorem (b) Using the Max Flow/Min Cut theorem or otherwise prove that given a collection (A i ) n 1 of sets and integers (d i ) n 1 one can find d i distinct representatives for A i if and only if for all S {1, 2, n} we have A i d i i S i S [To be precise we want to find distinct elements x ij for 1 i n, 1 j d i with x ij A i ] 3 June 2006, Question 10: Suppose that G = G(n) is a graph such that no two vertices of G are joined by 3 internally vertex disjoint paths Prove that e(g) 3 (n 1) 2 [Hint: Consider the block cut-vertex graph of G] 4 January 2006, Question 2: Prove that if G is a k-connected graph and S, T V (G) are disjoint subsets of vertices, each of size k then it is possible to find k disjoint paths P 1, P 2,, P k in G and labelings S = {s 1, s 2,, s k }, T = {t 1, t 2,, t k } such that P i is a path from s i to t i 5 June 2005, Question 3: Let k 1 Prove that if G is k-connected and S is a set of k vertices of G then there is a cycle C G with S V (C) [Hint: you may assume the Fan Lemma provided you state it clearly] 6 January 2005, Question 6: Let k be an integer with k 1, let G be a k-connected graph, and let S, T V (G) be disjoint sets of vertices of G, each of size at least k Prove that G contains k disjoint S, T paths (No version of the Fan Lemma may be assumed without proof) 7 January 2003, Question 9: 16

17 (a) Define a block of a graph G, and the block-cutvertex tree associated with G (b) Prove that if G has blocks B 1, B 2,, B k then n(g) = 1 k + k n(b i ) i=1 (c) Obtain, with justification, a formula for the number of spanning trees of G, given that the number of spanning trees of B i is t i 34 Coloring, Turán s Theorem 1 June 2008, Queston 7(b): Let G be a graph with 2m + 1 vertices and more than m (G) edges Prove that χ (G) > (G) 2 June 2008, Question 8: Prove that χ(g) = ω(g) when the complement of G is bipartite 3 June 2008, Question 10: [An alternative proof of Turán s theorem] Let G be a maximal K r -free graph (maximal in the sense of edges) on at least r 1 vertices (a) Prove that G contains a K r 1 (b) Show that if A is a clique in G of size r 1 then e(g) e(k r 1 ) + (n r + 1)(r 2) + e(g \ A) (c) Give a proof of Turán s theorem using the previous part 4 January 2007, Question 8: Let G be an edge-maximal graph on n vertices not containing a K r and suppose that n r +1 Let T r 1 (n) be the complete (r 1)-partite graph on n vertices whose class sizes are as equal as possible, and let t r 1 (n) = e(t r 1 (n)) (a) Prove that G contains a K r 1 on some subset A V (G) (b) Prove that e(g) ( ) r (r 2)(n r + 1) + e(g \ A) and deduce that e(g) t r 1 (n) (c) Prove that a graph on n vertices having t r 1 (n) edges and not containing a K r is T r 1 (n) 5 January 2007, Question 9: An interval graph is a graph whose vertices consist of closed intervals in R, and where two intervals are adjacent if they are not disjoint Prove that if G is an interval graph then χ(g) = ω(g) [Hint: consider using the greedy algorithm] 6 June 2007, Question 8: Let x be a vertex of a graph G For r 0 define G r = G[{y V (G) : d G (x, y) = r}] (In other words, G r is the subgraph of G induced by the vertices at distance r from x) Prove that χ(g) max r (χ(g r ) + χ(g r+1 )) 17

18 7 June 2006, Question 5: Let f(k) = k! ( ! + 1 3! ) + 1 k! k 3 s {}}{ Prove that R( 3, 3, 3 f(k), or in other words that if you color the edges of K n with k colors and n f(k) then the coloring contains a monochromatic triangle 8 June 2006, Question 8: State and prove Turán s theorem, including a proof that the extremal graph is unique up to isomorphism 9 June 2006, Question 9: Given an orientation of a graph G (an assignment of a direction to each edge of G) we define the length of the orientation to be the length of the longest directed path in G (a) Prove that if χ(g) k then G has an orientation of length at most k (b) Prove the converse: that if G has an orientation of length at most k then χ(g) k 10 June 2005, Question 2: Let G be a graph that does not contain two disjoint odd cycles Prove that χ(g) 5 Exhibit such a graph with χ(g) = 5 11 June 2005, Question 4 & January 2003, Question 7: State and prove Turán s theorem concerning the maximum number of edges in a graph on n vertices not containing a K r 12 January 2005, Question 8: Prove that χ(g) = ω(g) when G is bipartite 13 January 2005, Question 9 & June 2004, Question 10: (a) State Turán s theorem concerning the maximum number of edges in a graph on n vertices containing a K r (b) Prove that if G is a graph with n r + 1 vertices and t r 1 (n) + 1 edges then for every n with r n n there is a subgraph H of G with n vertices and at least t r 1 (n ) + 1 edges [Hint: consider a vertex in G of minimum degree] (c) From the previous part deduce Turán s theorem, and also the stronger fact that such a G contains two K r subgraphs sharing r 1 vertices 14 June 2003, Question 10: Define R(s, t) to be the minimum value of n such that every (non-necessarily) proper edge colouring of K n with the colours red and blue contains a monochormatic red K s or a monchromatic blue K t prove that for all s, t 2, R(s, t) R(s 1, t) + R(s, t 1) and deduce that for all such s, t ( ) s + t 2 R(s, t) s 1 18

19 15 June 2003, Question 8: State and prove Turán s Theorem concerning graphs not containing a copy of K r 16 January 2003, Question 8: (a) Prove the Szekeres-Wilf Theorem, stating that χ(g) 1 + max H G δ(h) (b) Given a set of N lines in the plane in general position (no two parallel, no three meeting at a point) define a graph on the vertex set consisting of the intersection points of the lines, with two vertices adjacent if they appear consecutively on one of the liens Prove that the chromatic number of this graph is at most 3 17 June 2002, Question 7: Let G be a graph containing no induced subgraph isomorphic to P 4 Prove that given any ordering of the vertices of G, the greedy algorithm colours G in ω(g) colours [Hint: If the greedy algorithm uses k colours, consider the smallest i such that G contains a clique consisting of vertices coloured i, i + 1, i 2,, k}] 35 Planar Graphs 1 June 2008, Question 9: Prove that a set of edges in a connected plane graph G forms a spanning tree of G if and only if the duals of the remaining edges form a spanning tree of the dual graph G 2 January 2006, Question 3: Prove that every planar graph G has χ(g) 5 3 June 2005, Question 5: (a) State Euler s formula concerning the number of faces, edges, and vertices of a plane graph (b) Prove that a plane graph with n vertices and e edges has e 3n 6 provided n 3 (c) An outerplane graph is a plane graph in which all the vertices are on the boundary of the outer face What is the maximum number of edges in an outerplane graph with n vertices? Justify your answer 4 January 2005, Question 10: Consider a connected plane graph G with dual G Prove that a subset of E(G) forms a spanning tree if and only if the duals of the remaining edges form a spanning tree of G 5 January 2002, Question 2: (a) Define the term maximal planar graph (b) Prove that if G is a maximal planar graph with p 3 vertices and q edges, then q 3p 6 (c) Prove that there exists only one 4-regular maximal planar graph 19

20 6 June 2002, Question 8: State and prove Euler s Formula concerning planer graphs The girth of a graph G is the length of the shortest cycle in G Prove that if G is planar and bridgeless with n vertices, m edges, and girth g, then 36 Hamiltonian Cycles 1 January 2007, Question 10: m g (n 2) g 2 (a) Let G be a graph on n 3 vertices such that for all pairs of non-adjacent vertices x, y we have d(x) + d(y) n Prove that G has a Hamilton cycle [Hing: consider a longest path in G] (b) Prove that the result in part (a) is best possible, by constructing a graph for every n 3, have n vertices and satisfying d(x) + d(y) n 1 for all pairs of non-adjacent vertices which does not have a Hamilton cycle 2 January 2002, Question 1: (a) Show that if a graph is not connected, then its complement is connected (b) A graph G has a proper edge coloring with k colors if no two edges of the same color meet at a common vertex Show that if G is a regular graph with degree 3, where G is Hamiltonian, then G has a proper edge coloring with three colors 20