Show that in a tournament (defined preceding mentioned theorem) it is always possible to rank the contestants so that the person ranked ith beats the person ranked (i +1)st. (Hint: Use the mentioned theorem.)

Show that a planar graph G with eight vertices and 13 edges cannot be 2-colored. (Hint: Use results in the said section to show that G must contain a triangle.)

A set of eight binary digits (0 or 1) are equally spaced about the edge of a disk. We want to choose the digits so that they form a circular sequence in which every subsequence of length three is different. Model this problem with a graph with four vertices, one for each different subsequence of two binary digits. Make a directed edge for each subsequence of three digits whose origin is the vertex with the first two digits of the edge’s subsequence and whose terminus is the vertex with the last two digits of the edge’s subsequence.

(a) Build this graph.

(b) Show how an Euler cycle (which exists for this graph) will correspond to the desired 8-digit circular sequence.

(c) Find such an 8-digit circular sequence with this graph model.

(d) Repeat the problem for 4-digit binary sequences.

(a) How many different Hamilton circuits are there in a complete graph on n vertices?

(b) Show that $K_n$, n prime greater than or equal to 3, can have its edges partitioned into 1/2 (n −1) disjoint Hamilton circuits.

(c) If 17 professors dine together at a circular table during a conference, and if each night each professor sits next to a pair of different professors, how many days can the conference last?

Consider a graph representing games played between a set of football teams with a directed edge from vertex A to vertex B if team A beats team B. Suppose that it is known that this graph has no directed circuits (i.e., no situation such as A beats B, B beats C, and C beats A). We can define a set of levels in this football graph as follows: A vertex with no outward edges (the team beat no other team) is at level 0; if all a vertex’s outward edges go to level-0 vertices (no-win teams), then the vertex is at level 1; and in general, a vertex is at level k if the greatest level of a team it beat is level k −1. Show that the level number of each vertex is a proper “coloring” of the vertices.

Prove that for any positive integer k, there exists a triangle-free graph G with $\chi(G)=k$.

(a) The proof should be by induction on k, the chromatic number. Initially for k = 3, we use the graph $G_3$ consisting of a 5-circuit.

(b) Assuming one can construct $G_k$ , a triangle-free graph with $\chi(G_k)=k$, one constructs $G_{k+1}$ by making k copies of $G_k$ and then adding $(n_k )^k$ vertices, where $n_k$ is the number of vertices in $G_k$ . Each new vertex has as its set of neighbors a different k-tuple consisting of one vertex from each copy of $G_k$. Confirm that this new graph is the desired $G_{k+1}$.

Suppose we are given an undirected connected graph representing a network of two-way streets.

(a) Show that there always exists a tour of the network in which a person drives along each side of every street once.

(b) Show that the tour in part (a) can be generated by the following rule: at any intersection, do not leave by the street first used to reach this intersection unless all other streets from the intersection have been used.

Consider 27 little cubes arranged in a 3 by 3 by 3 array (as in Rubik’s Cube). Form an associated graph with 27 vertices, one for each little cube, and with two vertices adjacent if they have touching faces (not just edges). Does this graph have a Hamilton path starting at the vertex corresponding to the middle inside cube and ending at one of the vertices corresponding to a corner cube?

Which of the following pairs of tours in the earlier figure can be combined without violating the 3-colorability requirement of the tour graph in the said example?

(a) Tours D and E

(b) Tours C and D

Show that a graph with at most two odd-length circuits can be 3-colored.

(a) Describe how to construct a circuit including all squares of an n by n chessboard, n even, using a rook. Then do the same using a king.

(b) Repeat part (a) for n odd.

Consider the following algorithm due to Fleury for building an Euler cycle, when one exists, in a single pass through a graph (without later adding side cycles as in the proof of the theorem). Starting at a chosen vertex a, build a cycle and erase edges after they are used (also erase vertices when they become isolated points). The one rule to follow in the cycle building is that one never chooses an edge whose erasure will disconnect the resulting graph of remaining edges.

(a) Apply this algorithm to build Euler cycles for the graph in

(i) The earlier figure

(ii) The mentioned figure (including deadheading edges)

(b) Prove that this algorithm works.

(c) Does this algorithm work for Euler trails? Explain.

Suppose a classroom has 25 students seated in desks in a square 5 by 5 array. The teacher wants to alter the seating by having every student move to an adjacent seat (just ahead, just behind, on the left, or on the right). Show that such a move is impossible.

A banquet center has eight different special rooms. Each banquet requires some subset of these eight rooms. Suppose that there are 12 evening banquets that we wish to schedule in a given week (seven days). Two banquets that are scheduled on the same evening must use different special rooms. Model and restate this scheduling problem as a graph-coloring problem.