Tag Archives: combinatorics

Generating Functions

In this post we will discuss generating functions. Generating functions are a powerful tool which can be used to solve many problems both in combinatorics and also in analysis.

Let R^\infty denote the set of all sequences of real numbers. For all i\ge 1 we let x^i denote the sequence which has value 1 for its (i+1)th term and all its other terms are zero. The symbol 1 stands for the sequence (1,0,0,\cdots). Also for any real number \alpha we define the product of \alpha and a sequence \alpha(a_0,a_1,\cdots) as (\alpha a_0,\alpha a_1,\cdots). We let two sequences {a_n} and {b_n} be equal if a_i=b_i for all i. We define the sum of {a_n} and {b_n} by the sequence {a_n+b_n} and the product by the sequence {c_n} where c_i=\sum_{j=1}^ia_jb_{i-j}. Clearly the sequence (a_0,a_1,\cdots ) is equal to the sequence a_0+a_1x+a_2x^2\cdots which we will also denote simply by \sum a_ix^i. Note that a_0 here stands for the sequence obtained as a product of a_0 and 1, i.e. a_0(1,0,0\cdots). Algebraically speaking \mathbb {R}^\infty, equipped with these operations, is an \mathbb {R}-algebra.

More importantly there is an analytic viewpoint of \mathbb {R}^\infty also. Readers who are familiar with the theory of power series can consider the elements of \mathbb {R}^\infty to be power series, i.e. each element is basically a function with its domain an open interval (or simply \{0\}). By standard theorems in analysis, if \sum a_ix^i and \sum b_ix^i both converge for |x|0 then for all such x, \sum a_ix^i=\sum b_ix^i if and only if a_i=b_i for all $i$. Hence the approach of considering \sum a_ix^i as a purely formal object may be considered equivalent to considering it as a power series.

However, we will soon see as to why convergence issues do not play any role in our context as long as the power series converges for at least one non zero x and there is more value in interpreting \sum a_ix^i as simply an element of \mathbb {R}^\infty.

Definition:
Let (a_n) be a real sequence such that the power series \sum a_ix^i converges for at least one non zero x. Then the function f which sends such an x to the power series \sum a_ix^i is called the generating function of the sequence. We frequently abuse notation and refer to the value of the generating function at some non-zero point x as the generating function (which is akin to referring the function f as f(x)).

Example:
Let (a_n) be the constant sequence of one’s. It is well known that for any real number x, if |x|<1 the series 1+x+x^2+\cdots converges to 1/(1-x). So the generating function of (a_n) is f where f(x)=1/(1-x) for all x at which the series converges.

The reason for requiring convergence at a non zero point is as follows. As soon as we have convergence at a non zero x, by a theorem of analysis it follows that there is convergence in an open interval around 0. Now, f has a unique power series expansion in that interval and so we are guaranteed that there is a one-one correspondence between the purely discrete object (1,1,\cdots) thought of as an element of \mathbb{R}^\infty and the generating function f. This can be exploited in the reverse direction, for if we wish to recover our sequence from the function f, then since f is defined by f(x)=1/(1-x)=1+x+x^2+\cdots x^n there is absolutely no ambiguity, and we cannot get back any other sequence. In fact, we may say that our sequence has been encoded within the definition of f, as a closed form expression 1/(1-x) .

If convergence was only given at 0, then such a one-one correspondence is not possible, since any closed form analytic function f, which is a_0 at 0 would become the generating function to the sequence (a_0,a_1,a_2,\cdots). So for any sequence (a_0,a_1,\cdots) we will consider its generating function to be defined by a_0+a_1x+a_2x^2\cdots as long as there is a non zero $x$ for which there is convergence, and once we have done that will not bother about any convergence issues at all.

The reader may be wondering what was the point of giving an algebraic approach initially, for a generating function really seems to have to do more with a power series. Furthermore in the notation of the first paragraph when we were considering a sequence as an element of \mathbb{R}^\infty we gave found that in our algebra (a_0,a_1,\cdots) is nothing but a_0+a_1x+a_2x^2+\cdots. This was not a power series but simply our notation for a sequence. It may appear confusing to have the same notation for two different objects, but it has been deliberately adopted for a very good reason. During our computations, we often manipulate our series so that we may no longer be sure whether convergence of a given series at a non-zero point is guaranteed. This poses a mathematical problem of the validity of our operations. However our ambiguous notation comes to our rescue at that instant, for what we really are doing at that point, without explicitly saying so, is not dealing with the series a_0+a_1x+a_2x^2+\cdots. Instead we are manipulating the sequence a_0+a_1x+a_2x^2+\cdots=(a_0,a_1,a_2,\cdots) with which there are absolutely no concepts of convergence attached. Of course if we need closed form expressions or some other analytic properties have to be established we need convergence so that one can use the one-one correspondence between the sequence and the generating function and dive in the world of analysis. In this way, a constant interplay between the discrete world of sequences and the analytic world of sequences brings out the real power of generating functions.

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The Inclusion Exclusion Principle

The Inclusion Exclusion Principle is one of most fundamental results in combinatorics. It deals with counting through a seive technique: i.e. we first overcount the quantity to be enumerated, then we try to balance out the overcounting by subtracting the extra. We may or may not subtract more then what is needed and so we count again the extra bits. Continuing in this way we “converge” at the correct count.

An easy example is |A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C| where A,B,C are three finite sets.

Here we start off by summing the number of elements in A,B,C seperately. The overcount is compensated by subtracting the number of elements in |A\cap B|,|A\cap C|, |B\cap C|. We actually compensate for a bit more then needed and so to arrive at the correct count we must add |A\cap B\cap C|.

Our goal here is to prove the inclusion-exclusion principle and then to look at a few corollaries. We first establish two lemmas.

Lemma 1: If X is any set containing n elements and k is any field, then the set of all functions f:X\to k is an n-dimensional vector space V over k with the naturally induced operations.

Proof: Let X=\{x_1,\cdots,x_n\}. It is easy to see that V together with the operations f+g, defined by (f+g)(x_i)=f(x_i)+g(x_i) and (\alpha f)(x_i)=\alpha (f(x_i)) for all x_i\in X is a vector space.

We now exhibit a basis for V consisting of n elements. For all x_i\in X let f_{x_i} be the function defined as f_{x_i}(y)=\begin{cases}         \hfill 1    \hfill & \text{ if }y=x_i \\        \hfill 0 \hfill & \text{ otherwise} \\    \end{cases}. Now, we claim that B=\{f_{x_i}:x_i\in X\} is a basis. Clearly it is spanning as for any f\in V we have f=\sum_{x_i\in X}f(x_i)f_{x_i}. It is linearly independent as \sum_{x_i\in X} \alpha_i f_{x_i}=0 means that for any j, with 1\le j\le n, we have \sum_{x_i\in X} \alpha_i f_{x_i}(x_j)=0, i.e. \alpha_j=0. This completes the proof.\Box

Lemma 2: If m\ge 0 then \sum_{i \mathop = 0}^m \left({-1}\right)^i \binom m i = \delta_{0m}.

Proof: If m>0 put x=1, y=-1 in the binomial theorem (x+y)^m=\sum_{i=0}^mx^{m-i}y^i. If m=0, then the sum on the left reduces to only one term: (-1)^0\binom 0 0. This is clearly 1.\Box

We now prove the Inclusion Exclusion principle. This theorem in its purest form, is simply a formula for an inverse of a linear operator. The theorem is as follows:

Theorem 3: Let S be a set with n elements. Let V be the 2^n dimensional vector space of all functions f:\mathcal{P}(S)\to k, where k is some field and \mathcal{P}(S) is the power set of S. Let \phi:V\to V be the linear operator defined by

\displaystyle\phi f(T)=\sum_{Y\supseteq T}f(Y)\text{ for all } T\subseteq S.

Then \phi^{-1} exists and is given by:

\displaystyle\phi^{-1} f(T)=\sum_{Y\supseteq T}(-1)^{|Y-T|}f(Y)\text{ for all } T\subseteq S.

Proof: To show \phi^{-1} as given above is indeed the inverse it suffices to show that \phi^{-1}\phi(f)=f for all f\in V.

Let f\in V. Then,

\displaystyle\phi^{-1}\phi f(T)=\sum_{Y\supseteq T}(-1)^{|Y-T|}\phi f(Y)=\sum_{Y\supseteq T}(-1)^{|Y-T|}\sum_{Z\supseteq Y}f(Z)=\sum_{Z\supseteq T}\Big(\sum_{Z\supseteq Y\supseteq T}(-1)^{|Y-T|}\Big)f(Z)

Now fix Z,T and let m=|Z-T|. Consider \displaystyle\sum_{Z\supseteq Y\supseteq T}(-1)^{|Y-T|}. Any Y is obtained by choosing some elements out of Z-T which has m elements, and taking the union of such elements with T. So for every i, with 0\le i\le m, there are exactly \binom m i ways of choosing a Y, which has i+|T| elements. Any such Y also has i elements in |Y-T|. So \displaystyle\sum_{Z\supseteq Y\supseteq T}(-1)^{|Y-T|}=\sum_{i=0}^m(-1)^i\binom m i=\delta_{0m}. This when substituted in the expression for \phi^{-1}\phi f(T) shows that \phi^{-1}\phi f(T)=f(T), which proves the theorem.\Box

We now discuss some corollaries of the Inclusion Exclusion principle. Let S be a set of properties that elements of a given set A may or may not have. For any T\subseteq S, let A'_T\subseteq A be those elements which have exactly the properties in T and no others. We define a function f_=:S\to \mathbb{R} such that f_=(T)=|A'_T|. Similarly, for any T\subseteq S, let A''_T\subseteq A be those elements which have at least the properties in T. We define a function f_\ge:S\to \mathbb{R} such that f_=(T)=|A''_T|. It is clear that \displaystyle f_\ge(T)=\sum_{Y\supseteq T}f_=(T) for any T\subseteq S, and so by the Inclusion Exclusion principle we conclude that

Corollary 4: For any T\subseteq S, we have \displaystyle f_=(T)=\sum_{Y\supseteq T}(-1)^{|Y-T|}f_\ge (Y).

In particular we have \displaystyle f_=(\emptyset)=\sum_{Y\subseteq S}(-1)^{|Y|}f_\ge (Y), which gives us a formula for the number of elements having none of the properties.\Box

In the above corollary we think of f_\ge(T) as the first approximation to f_=(T). So we “include” that much “quantity” in our initial count. From this we subtract all terms of the type f_=(Y) where Y has just one extra element then T. Thus we “exclude” that much “quantity” from our count. This gives a better approximation. Next we add all terms of the type f_=(Y) where Y has two extra elements then T, and so on. This is the reason behind the terminology inclusion-exclusion.

We now discuss another corollary. Let A be a finite set and let A_1,\cdots A_n be some of its subsets. We define a set of properties S=\{P_1,\cdots,P_n\} which elements of A may or may not enjoy as follows: For any i, with 1\le i\le n, x\in A satisfies the property P_i if and only if x\in A_i. Also for any I\subseteq [n], let \emptyset\ne A_I=\cap_{i\in I}A_i be the set of elements which have at least the properties P_i for i\in I. Define A_\emptyset to be A. By Corollary 4, \displaystyle f_=(\emptyset)=\sum_{I'\subseteq S}(-1)^{|I'|}f_\ge (I')=\sum_{I\subseteq [n]}(-1)^{|I|}|A_I| where in the second equality we correspond each subset of properties with a subset of [n]. We summarize this as

Corollary 5: Let A be a finite set and let A_1,\cdots A_n be some of its subsets. For any \emptyset\ne I\subseteq [n], let A_I=\cap_{i\in I}A_i and let A_\emptyset=A. The number of elements in A-\cup_{i=1}^n A_i is given by \displaystyle\sum_{I\subseteq [n]}(-1)^{|I|}|A_I|=|A|+\sum_{\emptyset\ne I\subseteq [n]}(-1)^{|I|}|A_I|.\Box

A further special case is obtained by considering any finite sets A_1,A_2,\cdots,A_n and letting A=\cup_{i=1}^nA_i. Then the above corollary translates to \sum_{I\subseteq [n]}(-1)^{|I|}|A_I|=0. Considering the case of I=\emptyset seperately, we see that |A|+\displaystyle\sum_{\emptyset\ne I\subseteq [n]}(-1)^{|I|}|\cap_{i\in I}A_i|=0. This easily yields the following corollary.

Corollary 6: Let A_1,A_2\cdots,A_n be any finite sets. For any \emptyset\ne I\subseteq [n], let A_I=\cap_{i\in I}A_i and let A_\emptyset=A. Then \displaystyle|A_1\cup A_2\cup\cdots\cup A_n|=\displaystyle\sum_{\emptyset\ne I\subseteq [n]}(-1)^{|I|-1}|\cap_{i\in I}A_i|.\Box

Now by grouping terms involving the same size of I we can restate both Corollary 5 and 6 as follows.

Corollary 7: Let A be a finite set and let A_1,\cdots A_n be some of its subsets. The number of elements in A-\cup_{i=1}^n A_i is given by
\displaystyle |A|+\sum_{k=1}^n(-1)^{k}\sum_{1\le i_1 < i_2 < \cdots <i_k\le n}|A_{i_1}\cap A_{i_2}\cap\cdots\cap A_{i_k}|.\Box

Corollary 8: If A_1,A_2\cdots,A_n are any finite sets then

\displaystyle|A_1\cup A_2\cup\cdots\cup A_n|=\sum_{k=1}^n(-1)^{k-1}\sum_{1\le i_1< i_2<\cdots<i_k\le n}|A_{i_1}\cap A_{i_2}\cap\cdots\cap A_{i_k}|.\Box

Corollaries 5 to 8 are often referred to as the principle of inclusion-exclusion themselves as in combinatorial settings they are the ones most often used. A further simplified version can also be derived from them when the intersection of any k distinct sets A_i always has the same cardinality N_k. In that case we only need to multiply N_k with the number of such ways to select the k sets to get the value of the inner sums in Corollaries 7 and 8.

Corollary 9: Let A be a finite set and let A_1,\cdots A_n be some of its subsets. Suppose that for any k with 1\le k\le n there exists a natural number N_k so that for any \{i_1,\cdots, i_k\}\subseteq [n] we have A_{i_1}\cap\cdots\cap A_{i_k}=N_k. Then the number of elements in A-\cup_{i=1}^n A_i is given by |A|+\sum_{k=1}^n(-1)^k\binom{n}{k}N_k and |A_1\cup A_2\cup\cdots\cup A_n|=\sum_{k=1}^n(-1)^{k-1}\binom{n}{k}N_k.\Box

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Graph Automorphisms

This post is concerning automorphisms of graphs, which quantify the symmetry existing within the graph structure. Given two graphs G and H, a bijection f:V(G)\to V(H) which maintains adjacency, i.e. xy\in E(G)\Leftrightarrow f(x)f(y)\in E(H), is called an isomorphism and the graphs G and H are called isomorphic. Clearly isomorphic graphs are essentially the same, with the superficial difference between them on account of different notation used in defining the vertex set. A isomorphism from the graph G to itself is called an automorphism. It is easy to see that the set of all automorphisms on a graph G together with the operation of composition of functions forms a group. This group is called the automorphism group of the graph, and is denoted by A(G).

In the remainder of this post we investigate some well known graphs and find out their automorphism groups.

The first graph we take up is the complete graph K_n. Any permutation of its n vertices is in fact an automorphism for adjacency is never lost. Its automorphism group is therefore S_n.

The next graph is the complete bipartite graph K_{m,n}. First consider the case m\not= n. The m vertices in the first partite set can be permuted in m! ways and similarly n! ways for the second partite set. Corresponding to each of these m!n! limited permutations we get automorphisms because adjacency is never disturbed. On the other hand, no automorphism can result from swapping a vertex from the first partite set and the second partite set because unless such a swap is done in its entirety (i.e. all the vertices from the first partite set swap places with the vertices in the second partite set), adjacency will be lost. A swap can be done in entirety only if m=n which is not the case we are considering. Hence no further automorphisms can result. Moreover by the multiplication rule it is simple to observe that the automorphism group would be isomorphic to S_m\times S_n.

In the case of m=n, we first pair off the vertices in the two partite sets against each other. This is also an automorphism, say f. Now for each of the m!m! ways of permuting vertices within partite sets, an additional automorphism arises. It is obtained in this fashion: After permuting the vertices within the partite sets by the particular way we swap each vertex with its pair in the other partite set. Clearly this yields 2m!m! automorphisms and furthermore no more are possible. Since every element of A(K_{m,m}) can be written as a unique product of an automorphism collection of the type covered in counting the first m!^2 ways (which is not hard to see is a normal subgroup, being of index 2) and of the subgroup \{Id,f\} so we see that the automorphism group is S_m\times S_m\rtimes \mathbb{Z}_2.

The next graph we take up is the cycle graph C_n. Firstly note that any automorphism can be obtained in this way: A given vertex v may be mapped to any of the n vertices available (including itself). As soon as that is done, an adjacent vertex to v has only two choices left: it can either be in the counter clockwise direction to v or in the clockwise direction to v. Once that choice is also made, no other choices are required. Hence we get 2n automorphisms this way and there can be no others. Also, it is clear that two kinds of automorphisms suffice to generate this group: rotation, and swapping the notion of clockwise and counter clockwise (assuming we draw the cycle graph as equally spaced points on the unit circle; there is no loss of generality in doing that). But both these automorphisms also generate the dihedral group D_n which also has 2n elements. It follows that A(C_n)=D_n.

The final graph we take up is the well known Petersen graph. Instead of directly considering what possible functions are there in its automorphism group (although such an approach is possible) we approach the problem through the concept of line graphs.

Definition: A line graph L(G) of a graph G is the graph whose vertices are in one to one correspondence with the edges of G, two vertices of L(G) being adjacent if and only if the corresponding edges of G are adjacent.

Lemma 1: L(K_5) is the complement of the Petersen graph.
Proof: It is clear that if the vertices of K_5 are labelled 1,2,\ldots,5 then its 10 edges are the {5 \choose 2} 2-subsets of \{1,\cdots,5\}. The line graph L(K_5) thus has 10 vertices, labeled by these 10 2-subsets \{i,j\}. Two vertices \{i,j\}, \{k,\ell\} are adjacent in L(K_5) iff the two 2-subsets have a nontrivial overlap. The complement of L(K_5) is the graph with the same 10 vertices, and with two vertices being adjacent iff the corresponding two 2-subsets are disjoint. But this is the very definition of the Petersen graph.\Box

Lemma 2: A(G) is equal to A(\bar{G}).
Proof: If \sigma\in A(G) then for any two vertices x,y we have xy\in E(G)\Leftrightarrow \sigma(x)\sigma(y)\in E(G), i.e. xy\not\in E(G)\Leftrightarrow \sigma(x)\sigma(y)\not\in E(G), i.e. xy\in E(\bar{G})\Leftrightarrow \sigma(x)\sigma(y)\in E(\bar{G}) so that \sigma\in A(\bar{G}). The reverse implication follows by replacing G by \bar{G}. \Box

Theorem 3: The automorphism group of the Petersen graph is S_5.
Proof: In view of Lemma 1 and 2 it suffices to find out A(L(K_5)) for the automorphism group of the Petersen graph is going to be the same. We let K_5 have the vertex set \{1,\cdots,5\} in the sequel.

Take any automorphism f of K_5. If we have two edges ab,cd\in K_5 with f(a)f(b)=f(c)f(d), then either of two cases arise. Either f(a)=f(c) or not. If f(a)=f(c) then obviously f(b)=f(d) and so by injectivity of f we have ab=cd. If f(a)\not=f(c) then it must be that f(a)=f(d). This means that f(b)=f(c) and again by injectivity we have ab=cd. What this means is that the function induced by f on E(K_5) in the natural way is injective. It is also surjective as for any xy\in E(K_5) clearly f\{f^{-1}xf^{-1}y\}=xy. Finally, this function is an automorphism since \{xy,xz\}\in E(L(K_5)) clearly implies and is implied by \{f(x)f(y),f(x)f(z)\}\in E(L(K_5)) as there is a common vertex. As our definition of the induced function is obtained in a definite way we have shown that every automorphism of K_5 induces a unique automorphism of L(K_5). Moreover, it is easy to see that if f_1,f_2 are two automorphisms then the automorphism induced by f_1\circ f_2 is the same as the automorphism induced by f_1 composed by f_2.

We now show that given an automorphism of L(K_5) we can obtain an automorphism of K_5 which induces it in the natural way. Let \pi\in A(L(K_5)). It is easy to see that the 4-cliques of L(K_5) originate from the stars K_{1,4} of K_5. So L(K_5) has exactly 5 4-cliques, say C_1,\cdots ,C_5 where C_i contains 4 vertices corresponding to the 4 edges in K_5 that are incident to a vertex i in K_5. Since \pi is an automorphism it sends 4-cliques to 4-cliques. Also, \pi must send two different 4-cliques C_i,C_j with i\ne j to different 4-cliques, because if it sends them to the same 4-clique then a collection of at least 5 vertices is mapped to a collection of 4 vertices, a contradiction to the injectivity of \pi. So \pi induces a permutation of the C_i‘s.

Now suppose \pi and \pi' are two different automorphisms in A(L(K_5)). Then they differ on at least vertex in L(K_5), say on the vertex ij\in E(K_5). Now given any vertex xy in L(K_5) consider the intersection of the 4-cliques C_x and C_y. If pq is some vertex in C_x\cap C_y then pq as an edge in K_5 is part of stars with centers x and y, i.e. pq=xy. Hence the intersection contains only the vertex xy. Every vertex of L(K_5) arises in this way. So if \pi(ij)\ne \pi'(ij), then either \pi (C_i)\ne \pi'(C_i) or \pi(C_j)\ne \pi'(C_j) for otherwise \pi (C_i\cap C_j)=\pi'(C_i\cap C_j).

Hence every automorphism of L(K_5) induces a unique permutation of the C_i‘s. Moreover distinct automorphisms induce distinct permutations so that the automorphisms and the permutations can be put in one-one correspondence. Consider an automorphism f of the vertices of K_5 where f(i)=j if C_i\to C_j in the permutation corresponding to \pi. Now a vertex \{x,y\}\in E(K_5) of L(K_5). This is also the intersection of the 4-cliques C_x and C_y and so \pi(\{x,y\})=\pi(C_x\cap C_y)=\pi(C_x)\cap\pi (C_y)=C_{f(x)}\cap C_{f(y)}=f(x)f(y). This shows that f induces \pi as an automorphism.

Hence we have shown that A(K_5)\equiv A(L(K_5)). So the Petersen graph has the automorphism group S_5. \Box

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Bipartite graphs

The topic of this post is bipartite graphs. These are graphs whose vertex set can be partitioned into two parts, called partite sets, such that all edges xy of the graph are such that x and y are in different partite sets. The most common examples of bipartite graphs are the trees and even cycles. Any union of bipartite graphs obviously yields another bipartite graph. The complete bipartite graph K_{m,n} consists of two partite sets X and Y containing m and n elements respectively with all possible edges between X and Y filled out.

To conclude here is a list of characterizations.

Suppose G is a graph with at least two vertices. The following are equivalent:

1. G is bipartite.

2. Every cycle of G is even.

3. \chi(G)=2.

Proof:
1 \Leftrightarrow 2: Suppose G is bipartite and there is a cycle v_1v_2\cdots v_kv_1 which has an odd number of edges. Now v_i for odd i is in the same partite set as v_1. But since the cycle is odd v_1 and v_k are in the same partite set which is a contradiction.

Conversely suppose G has no odd cycles and x\in V(G). Let X=\{y\in V(G):d(x,y) is even \} and Y=\{y\in V(G):d(x,y) is odd \}. Then X and Y partition V(G). Suppose pq\in E(G) and p,q\in X. Neither p nor q is x for otherwise d(x,q) (resp d(x,p)) is odd. So it makes sense to talk of a path from x to p (resp q). Let x=v_0v_1\cdots v_{2k-1}v_{2k}=p and x=w_0w_1\cdots w_{2m-1}w_{2m}=q be the shortest such paths. (Note that they are necessarily even). Let the last common vertex in the two paths be v_i=w_i. Now the cycle v_iv_{i+1}\cdots v_{2k}w_{2m}w_{2m-1}\cdots w_{i+1}w_i=v_i is an odd cycle. This contradiction proves there are no edges within vertices of X. Similarly there are no edges within vertices of Y. The result follows.

1 \Leftrightarrow 3: This is more of a restatement of the definition of a bipartite graph then a characterization, since the two partite sets may be considered as differently colored subsets of V(G).\Box

In addition to the above general characterizations the following are also true:

a) G is bipartite iff the spectrum of G is symmetric with respect to the origin.

b) Suppose G is connected and \lambda is its maximum eigenvalue. Then G is bipartite iff -\lambda is an eigenvalue of G.

c) Suppose G is planar. Then G is bipartite iff b(R), the bound degree of a region R, is even for every region R.

I will defer a proof of these statements.

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On unabridged versions of Ramsey’s theorem

In continuation of our discussions on Ramsey theory in this post we plan to prove the unabridged versions of Ramsey’s theorem. While the abridged versions dealt with graphs, unabridged versions deals with hypergraphs. A hypergraph, is a set of vertices V together with a set of subsets of V of a fixed order n. If n=2 then we get back the ordinary definition of a graph.

An example of a hypergraph of order 3 is given below:

The vertex set V=\{a,b,c,d,e\} and the hyperedges are given by: \{a,b,c\}, \{b,d,e\} and \{a,b,d\}.

The infinite unabridged version of Ramsey’s theorem may now be described as follows: Given a complete infinite hypergraph of order n with vertex set \mathbb{N}, and a r-coloring of its hyperedges we are guaranteed a infinite complete monochromatic subgraph.

Let us try to express this more succinctly. We will extend the previously defined arrows notation as follows: The notation \alpha\rightarrow (\beta)^n_r will mean that every r-colored complete hypergraph of order n on \alpha vertices contains a monochromatic complete subgraph with \beta vertices. Since hypergraphs do not seem to be as picturesque as graphs, so maybe the idea will be better expressed in terms of sets: \alpha\rightarrow (\beta)^n_r equivalently means that for any assignment of r-colors to the n-subsets of \alpha, there is a particular color (say red) and a subset X of \alpha of size \beta such that all n-subsets of X are red. (Note that \alpha,\beta etc can be any cardinal numbers.)

With the notation that \omega stands for the cardinality of \mathbb{N} the infinite version of Ramsey’s theorem now is:

Theorem: \forall n,r\in\mathbb{N}, \omega\rightarrow(\omega)^n_r.

The proof is essentially on the same lines as in the abridged case, with the addition that we induct on n. (n may intuitively be thought of as the “hyper-ness” of our hypergraph.)

Proof: By induction on n. If n=1 we just get the infinite pigeonhole principle: Any finite partitioning of \mathbb{N} contains an infinite part. Indeed, both the infinite and finite Ramsey theorems may be thought of as gigantic generalizations of the pigeonhole principle.

Now suppose that for a fixed n,r we have \omega\rightarrow(\omega)^n_r. Now consider any r-coloring f of the n+1-subsets of \omega. Just like in the proof of the abridged case we build a sequence (w_k). In the abridged case we applied the infinite pigeonhole principle repeatedly at this point. Here we will use the induction hypothesis repeatedly. Let w_1=1. Consider the r-coloring f_{w_1} of all n-subsets of \omega-\{1\} given by f_{w_1}(X)=f(\{1\}\cup X). By the induction hypothesis there will be a countable subset of \omega whose all n-subsets will be monochromatic. Call that subset V_1. Let the least element of V_1 be designated as w_2.

The construction of (w_k) follows by induction: If the set V_k and its least element w_{k+1} have been found, consider the r-coloring f_{w_{k+1}} of the n-subsets of V_k given by f_{w_{k+1}}(X)=f(\{w_{k+1}\}\cup X). By the induction hypothesis it contains a countable set all of whose n-subsets are monochromatic. Call that subset V_{k+1}. Its least element is designated as w_{k+2}.

The sequence (w_k) has the property that given any w_i the union of \{w_i\} with any n-order subset containing elements from w_{i+1},w_{i+2},w_{i+3}\cdots has the same color (with respect to the the original coloring f). Let us call w_i as c_j-based if this color is c_j. The fact that all the members of the sequence (w_k) are based in any of r-ways induces a partitioning of the sequence (w_k). By the infinite pigeonhole principle we are guaranteed an infinite part. Now, by ordering the terms appropriately in this infinite part we have a subsequence (w_{k_j}) of (w_k) in which all vertices are the same color base, say c_1. It is now easy to observe that all the n+1-subsets of this infinite part are colored c_1, which proves the theorem.\Box

We now turn to the finite version. In our arrows notation, the natural generalization from the abridged analogue reads:

Theorem: \forall l,n,r\in\mathbb{N}, \exists m\in\mathbb{N} such that m\rightarrow (l)_r^n.

Proof: The proof is identical to the abridged case with cosmetic changes. By way of contradiction assume that there is an l such that for all m\in\mathbb{N} we have m\nrightarrow (l)_r^n. We will use the notation \hat{K_i} for a hypergraph on i vertices where all possible n-subsets of the vertices are the hyperedges. Also let G be a hypergraph with vertices V=\{v_i:i\in\mathbb{N}\} and let the hyperedges of G be enumerated by E=\{E_i:E_i\subset\mathbb{N}, |E_i|=n\}. We construct a (rooted) tree T as follows:

1. First introduce a root vertex rt.

2. Each vertex is allowed to have at most r children which correspond to the r-colors, subject to it satisfying the criteria below. A child is always labeled by one among the r-colors. (Call the colors c_1,c_2\cdots c_r for convenience).

3. A child c_i is permitted if and only if its introduction creates a path of some finite length k starting from the root, so that if the hyperedges E_1,E_2\cdots E_k are colored by the colors used in the path in the same order, then the corresponding subgraph in G does not contain a monochromatic \hat{K_l}. For example if the introduction of a child c_i creates the k length path rt,c_a,c_b\cdots ,c_i and the hyperedges E_1,E_2,\cdots E_k when colored c_a,c_b\cdots c_i don’t contain a monochromatic \hat{K_l} the child c_i is permitted to be added to T.

Note that for all m, there always exists a coloring of \hat{K_m} such that no monochromatic \hat{K_l} exists within. So the situation that a child cannot be added to any vertex at a given level k cannot arise. For we can always take a coloring of \hat{K_{k+n}} containing no monochromatic \hat{K_l}. Since any k hyperedges in it would yield a sequence of colors already existing in T, we know which vertex to add the child to. We give the child the color corresponding to any other edge. Hence we can forever keep adding children and so T is infinite. It is also obvious that each level k of T has at most r^k vertices and so each level is finite.

Now by Konig’s lemma there will be an infinite path in T. This infinite path provides a r-coloring of G that contains no monochromatic \hat{K_i} and hence no monochromatic infinite hypergraph which contradicts the infinite Ramsey theorem proved above. This contradiction proves the theorem.\Box

The astute reader may have noted that we have used weak versions of the axiom of choice occasionally in the proof. Ramsey himself was aware of this and remarked so in his original paper. In fact, it can be proved that Ramsey’s theorem cannot ever be proved without using some form of the axiom of choice. Moreover it can be shown that a weak form of the axiom of choice is equivalent to Ramsey’s theorem. Hence Ramsey’s theorem may be interpreted as a choice axiom.

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Arrows, Ramsey numbers and the Party Problem

Here come the arrows! The notation \alpha\rightarrow (\beta)^2_r means that every r-colored complete graph on \alpha vertices contains a monochromatic complete subgraph with \beta vertices. Using the notation \omega for the cardinality of the natural numbers the abridged Ramsey theorems state that \omega\rightarrow (\omega)^2_r and that \forall n\exists m,m\rightarrow (n)^2_r.

Now let us take a particular case: n=3,r=2. Our objective here is to prove that 6\rightarrow (3)^2_2. In other words, if the edges of K_6 are 2-colored then there exists a monochromatic K_3. Fix a vertex v in K_6. From the 5 edges emanating from v, at least three must share a color: call it c. Now if all the edges in triangle made by those three vertices are not colored c, we are done for we have found a monochromatic K_3 in the other color. If even one edge, say xy among the triangle made by those three vertices is of color c then we have a monochromatic K_3 colored c with vertices x,y and v.

The fact that 6\rightarrow (3)^2_2 is also referred to as the solution of the party problem: Prove that in any gathering of six people there are three mutual friends, or three mutual strangers. By representing the people as vertices and their relationships via the colors on the edges connecting them we easily see that a monochromatic triangle is guaranteed and therefore the result holds. It is also obvious that the gathering may contain more then six people, a monochromatic triangle is still guaranteed within (in fact, it will only become more likely with more people). In other words: m\rightarrow (3)^2_2\forall m\ge 6.

Is six the least such number? Indeed it is as the following graph demonstrates:

We note that there is no monochromatic triangle in the above 2-colored K_5.

For a given n the least such number m,m\rightarrow (n)^2_r is called the diagonal Ramsey number R(n,n). While such numbers are guaranteed to exist by Ramsey’s theorem, actually finding them is hard indeed. It is trivial to see that R(1,1)=1 and R(2,2)=2. Our proof above illustrates that R(3,3)=6. It has also been proven that R(4,4)=18. However all efforts to find out the exact value of any other diagonal Ramsey number have failed. It is one of those problems, which are easily stated but extremely difficult to attack.

Regarding R(5,5), Joel Spencer comments in his book, Ten lectures on the Probabilistic Method:

Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5,5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6,6). In that case, he believes, we should attempt to destroy the aliens.

In closing it may be remarked that the Ramsey number R(m,n) is defined to be the least positive integer such that any 2-coloring of K_{R(m,n)} in two colors, say red and blue, contains either a red K_m or a blue K_n. This definition also has the natural generalization for more then two colors. Although the existence of all Ramsey numbers is guaranteed by Ramsey’s theorem, only few have been discovered so far.

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On abridged versions of Ramsey’s theorem

This post will aim to prove two abridged versions of Ramsey’s theorem, the result which gave its name to the area of mathematics now known as Ramsey theory. Ramsey theory may be understood as ramifications of a philosophical principal. The principle may be summarized as: “Complete disorder is an impossibility. Any structure will necessarily contain an orderly substructure.” Ramsey’s theorem is an example of this principle. Van der Waerden’s theorem is another example.

First a bit of history regarding this theorem. Frank Ramsey, a 25 year old British mathematician, proved this result in 1928 as a part of his investigations in mathematical logic. Unfortunately he passed away in 1930 (even before his theorem was formally published) and did not prove anything else related to this. In 1933, Skolem published another proof of the theorem, and then in 1934, Erdos and Szekeres found yet another. It was the Erdos and Szekeres paper which gave Ramsey’s theorem a wide audience. Further work by Erdos in this area led mathematicians to consider other Ramsey type theorems.

Here we consider abridged versions of Ramsey’s theorem which can be easily understood in the context of graphs. We will imitate Ramsey’s original approach (though not his proof) to first prove an infinite version (which deals with infinite graphs) and then proceed towards the finite analogue.

So what is the infinite version. Briefly it says that if we have an infinite complete graph on countable vertices and if we use r colors c_1,c_2\cdots c_r to arbitrarily color the edges then we are guaranteed an infinite monochromatic subgraph. Here our usage of coloring an edge is just a fanciful way to describe partitioning the edge set into finitely many parts. More formally, by a r-coloring of the edges of a graph G we mean any function with domain E(G) and range \{1,2\cdots r\} which induces such a partitioning. The colors are the labels that we may use for this partitioning.

Theorem: Let G be the complete infinite graph with vertices V=\{v_i:i\in\mathbb{N}\}. Given any r-coloring of the edges, G will contain an infinite complete monochromatic subgraph.

Ramsey’s original formulation was more general. He considered the set of vertices to be any infinite set (and not just countable). That is not much gained because a coloring of an uncountable vertex graph induces a coloring of a countable vertex subgraph. Within this countable subgraph we can find a monochromatic infinite complete graph, which is also a subgraph of the original. However he also generalized in another sense: Instead of coloring edges he considered colorings of hyperedges of a fixed order (an edge is a special case: it is a hyperedge of order 2). This too can be overcome by induction on the “hyper-ness” of the graph. We choose to postpone such discussions.

Proof: Suppose the edges of G have been colored using the colors c_1,c_2,\cdots c_r. We will build an infinite subsequence of the vertices in V by applying the infinite pigeonhole principle. Let w_1=v_1. Now w_1 is connected with every other vertex in G and the edges making this connection are colored in any of r-ways. We partition the set of all these vertices in this fashion: If v is connected to w_1 via an edge colored c_i we consider it to be a member of the set P_i. Clearly, \{P_i:i=1,2\cdots r\} is a partitioning of V-\{w_1\}. Now, since \cup P_i is infinite so one of the P_i must also be infinite. (Here we are using the infinite pigeonhole principle.) Note that there may be more then one infinite P_i. At any rate choose the “smallest” vertex among the infinite parts (or the vertex with the least i in \{v_i:i\in P_j, P_j is infinite \}. Rename that particular P_i as V_1 and that vertex as w_2.

At this point the construction of the sequence becomes obvious. If for some n we have already found a w_n and and a V_n then we can partition the vertices in V_n connected to w_n into r-parts depending on how they are connected with w_n. Choosing the smallest vertex among the infinite parts yields w_{n+1} and the part it belongs to yields V_{n+1}. Since V_{n+1} is infinite so we have no fear of running out of vertices.

Now note an important property of the sequence (w_n). Given any w_i its connections with w_{i+1},w_{i+2},w_{i+3}\cdots are all of the same color by definition. Let us call w_i as c_j-based if the edges w_iw_{i+1},w_iw_{i+2},w_iw_{i+3}\cdots are all colored c_j. The fact that all the members of the sequence (w_n) are based in any of r-ways induces a partitioning of the sequence (w_n). As before by the pigeonhole principle we are guaranteed an infinite part. Now, by ordering the terms appropriately in this infinite part we have a subsequence (w_{n_k}) of (w_n) in which all vertices are the same color base, say c_1. It is now easy to observe that the subgraph induced by these vertices has all edges colored c_1, which proves the theorem.\Box

We now turn our attention to the finite analogue. The infinite version guarantees an infinite monochromatic complete graph within an arbitrarily colored infinite countable graph (with countable vertices). The finite version guarantees a finite monochromatic complete graph of desired size within an arbitrarily colored finite countable graph (with suitable number of vertices). More formally the theorem states:

Theorem: Let r,l_1,l_2,\cdots l_r be natural numbers. Then there exists an m\in \mathbb{N} such that any r-coloring of K_m contains a monochromatic K_{l_i} for some i entirely colored in color i.

We will prove the theorem with l_1=l_2=\cdots =l_r=n. This is because no loss of generality occurs by assuming this. To see this suppose r=2,l_1=5 and l_2=7. Now if we have proved the theorem for r=2,l_1=l_2=7 we have also guaranteed the existence of an m such that any 2-coloring of K_m contains a K_7 colored in either of the two colors. Clearly the same m works for l_1=5,l_1=7 as a 1-colored K_7 would contain a 1-colored K_5.

Proof: Our modified claim is that for all r,n\in\mathbb{N} there is an m\in\mathbb{N} such that any r-coloring of K_m contains a monochromatic K_n. By way of contradiction assume that there is an n\in\mathbb{N} such that for every m\in \mathbb{N} there is a r-coloring of K_m containing no monochromatic K_n. Now let G be a complete graph with vertices V=\{v_i:i\in\mathbb{N}\} and let the edges of G be enumerated by E=\{e_i:i\in\mathbb{N}\}. We construct a (rooted) tree T as follows:

1. First introduce a root vertex rt.

2. Each vertex is allowed to have at most r children (a child of a vertex x is defined to be an adjacent vertex y so that d(rt,x)+1=d(rt,y)) which correspond to the r-colors, subject to it satisfying the criteria below. A child is always labeled by one among the r-colors. (Call the colors c_1,c_2\cdots c_r for convenience).

3. A child c_i is permitted if and only if its introduction creates a path of some finite length k starting from the root, so that if the edges e_1,e_2\cdots e_k are colored by the colors used in the path in the same order, then the corresponding subgraph in G does not contain a monochromatic K_n. For example if the introduction of a child c_i creates the k length path rt,c_a,c_b\cdots ,c_i and the edges e_1,e_2,\cdots e_k when colored c_a,c_b\cdots c_i don’t contain a monochromatic K_n the child c_i is permitted to be added to T.

Note that for all m, there always exists a coloring of K_m such that no monochromatic K_n exists within. So the situation that a child cannot be added to any vertex at a given level k cannot arise. For we can always take a coloring of K_k containing no monochromatic K_n. Since any k edges in it would yield a sequence of colors already existing in T, we know which vertex to add the child to. We give the child the color corresponding to any other edge. Hence we can forever keep adding children and so T is infinite. It is also obvious that each level k of T has at most r^k vertices and so each level is finite.

Now by Konig’s lemma there will be an infinite path in T. This infinite path provides a r-coloring of G that contains no monochromatic complete graph and hence no monochromatic infinite complete graph which contradicts the infinite Ramsey theorem proved above. This contradiction proves the theorem.\Box

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