For k ≥ 2, a graph G is said to be k-vertex connected, or simply k-connected, when |V(G)| ≥ k + 1 and the removal of any k − 1 vertices (and any incident edges) from G does not result in a disconnected graph. We use 1-connected as a synonym for connected.

If a graph G with at least k + 1 vertices is not k-connected and the deletion of a set S of k − 1 vertices disconnects it, there is a partition of V(G) \ S into nonempty sets X, Y with no edges crossing (one end in X, one in Y). Let H and K be the subgraphs induced by X ∪ S and Y ∪ S, except that edges with both ends in S are to be put in one and only one of H or K. Then we obtain edge-disjoint subgraphs H and K whose union is G and such that |V(H)∩V(K)| = k − 1. Conversely, if such subgraphs H and K exist and each contains at least one more vertex than their intersection, then G is not k-connected.

A graph is said to be nonseparable when it is 2-connected and has no loops, or when it is a bond-graph (with two vertices and any positive number of edges joining them, including the link-graph with one such edge), a loop-graph (one edge joining a single vertex to itself), or a vertex-graph. All polygons, for example, are nonseparable; path-graphs or other trees with at least two edges are not.

The favorable reception of our book and its use for a variety of courses on combinatorial mathematics at numerous colleges and universities has encouraged us to prepare this second edition. We have added new material and have updated references for this version. A number of typographical and other errors have been corrected. We had to change “this century” to “the last century” in several places.

The new material has, for the most part, been inserted into the chapters with the same titles as in the first edition. An exception is that the material of the later chapters on graph theory has been reorganized into four chapters rather than two. The added material includes, for example, discussion of the Lovàsz sieve, associative block designs, and list colorings of graphs.

Many new problems have been added, and we hope that this last change, in particular, will increase the value of the book as a text. We have decided not to attempt to indicate in the book the level of difficulty of the various problems, but remark again that this can vary greatly. The difficulty will often depend on the experience and background of the reader, and an instructor will need to decide which exercises are appropriate for his or her students. We like the idea of stating problems at the point in the text where they are most relevant, but have also added some problems at the end of the chapters. It is not true that the problems appearing later are necessarily more difficult than those at the beginning of a chapter.

We shall first look at a few so-called coloring problems for graphs.

A proper coloring of a graph G is a function from the vertices to a set C of ‘colors’ (e.g. C = {1, 2, 3, 4}) such that the ends of every edge have distinct colors. (So a graph with a loop will admit no proper colorings.) If |C| = k, we say that G is k-colored.

The chromatic number χ(G) of a graph G is the minimal number of colors for which a proper coloring exists.

If χ(G) = 2 (or χ(G) = 1, which is the case when and only when G has no edges), then G is called bipartite. A graph with no odd polygons (equivalently, no closed paths of odd length) is bipartite as the reader should verify.

The famous ‘Four Color Theorem’ (K. Appel and W. Haken, 1977) states that if G is planar, then χ(G) ≤ 4.

Clearly χ(Kn) = n. If k is odd then χ(Pk) = 3. In the following theorem, we show that, with the exception of these examples, the chromatic number is at most equal to the maximum degree (R. L. Brooks, 1941).

Theorem 3.1.Let d ≥ 3 and let G be a graph in which all vertices have degree ≤ d and such that Kd+1 is not a subgraph of G. Then χ(G) ≤ d.

Proof 1: As is the case in many theorems in combinatorial analysis, one can prove the theorem by assuming that it is not true, then considering a minimal counterexample (in this case a graph with the minimal number of vertices) and arriving at a contradiction.

The following problem originated in communication theory. For a telephone network, a connection between terminals A and B is established before messages flow in either direction. For a network of computers it is desirable to be able to send a message from A to B without B knowing that a message is on its way. The idea is to let the message be preceded by some ‘address’ of B such that at each node of the network a decision can be made concerning the direction in which the message should proceed.

A natural thing to try is to give each vertex of a graph G a binary address, say in {0, 1}k, in such a way that the distance of two vertices in the graph is equal to the so-called Hamming distance of the addresses, i.e. the number of places where the addresses differ. This is equivalent to regarding G as an induced subgraph of the hypercube Hk, which has V(Hk) ≔ {0, 1}k and where k-tuples are adjacent when they differ in exactly one coordinate. The example G = K3 already shows that this is impossible. We now introduce a new alphabet {0,1, ⋆} and form addresses by taking n-tuples from this alphabet. The distance between two addresses is defined to be the number of places where one has a 0 and the other a 1 (so stars do not contribute to the distance).

In this chapter we give an introduction to a large and important area of combinatorial theory which is known as design theory. The most general object that is studied in this theory is a so-called incidence structure. This is a triple S = (P, B, I), where:

1. P is a set, the elements of which are called points;

2. B is a set, the elements of which are called blocks;

3. I is an incidence relation between P and B (i.e. I ⊆ P × B). The elements of I are called flags.

If (p, B) ∈ I, then we say that point p and block B are incident. We allow two different blocks B1 and B2 to be incident with the same subset of points of P. In this case one speaks of ‘repeated blocks’. If this does not happen, then the design is called a simple design and we can then consider blocks as subsets of P. In fact, from now on we shall always do that, taking care to realize that different blocks are possibly the same subset of P. This allows us to replace the notation (p, B) ∈ I by p ∈ B, and we shall often say that point p is ‘in block B’ instead of incident with B.

It has become customary to denote the cardinality of P by v and the cardinality of B by b.

A graph G consists of a set V (or V(G)) of vertices, a set E (or E(G)) of edges, and a mapping associating to each edge e ∈ E(G) an unordered pair x, y of vertices called the endpoints (or simply the ends) of e. We say an edge is incident with its ends, and that it joins its ends. We allow x = y, in which case the edge is called a loop. A vertex is isolated when it is incident with no edges.

It is common to represent a graph by a drawing where we represent each vertex by a point in the plane, and represent edges by line segments or arcs joining some of the pairs of points. One can think e.g. of a network of roads between cities. A graph is called planar if it can be drawn in the plane such that no two edges (that is, the line segments or arcs representing the edges) cross. The topic of planarity will be dealt with in Chapter 33; we wish to deal with graphs more purely combinatorially for the present.

Thus a graph is described by a table such as the one in Fig. 1.1 that lists the ends of each edge. Here the graph we are describing has vertex set V = {x, y, z, w} and edge set E = {a, b, c, d, e, f, g}; a drawing of this graph may be found as Fig. 1.2(iv).