COURSE CODE: 1700107
EXAMINATION FOR GRAPH THEORY AND ITS APPLICATIONS
6.30 – 7.30 PM., NOV. 20, 2019Dear graph theory beginners,
All graphs mentioned in this examination are finite, simple, and have at least 3 vertices. A bonus problem, which values an extra 100 points, is provided for students who found the others trivial. Detailed scoring rules are on the last page for self-estimation of your score.
I (10 points) What is a simple graph?
II (30 points) Please choose a concept and a theorem from the following, state them precisely. 1. A spanning tree of a graph.
2. A k-factor of a graph, wherekis a positive integer. 3. The chromatic polynomial of a graph.
4. A planar graph. 5. An Eulerian graph. 6. A Hamiltonian graph.
7. Hall’s theorem on a characterization of the existence of a matching in a bipartite graph that saturates one part.
8. Tutte’s 1-factor theorem on the existence of a 1-factor in a graph. 9. Menger’s theorem on the cut size and the number of disjoint paths.
10. Tur´an’s theorem on the structure of graphs without (r+ 1)-cliques that maximizes the number of edges, whereris a positive integer.
11. Euler’s formula: an equality between the number of vertices, edges and faces in certain graphs.
III (30 points) Each problem in this section collects 4 statements. LetN be the number of correct statements in each of the collections. Please answer
• A ifN 6∈ {2,3,4};
• B ifN = 2;
• C if N= 3;
• D ifN = 4.
1. a) A simple graph is bipartite if and only if it has no odd cycles.
b) Any simple graph in which every vertex has degree 2 is the union of cycles. c) The complete graphK5 and the complete bipartite graphK3,3 cannot be drawn
without crossings on the sphere.
d) A simple graph is Hamiltonian if and only if its closure is Hamiltonian.
2. Denote by P the Petersen graph. a) P is bipartite.
b) The girth ofP is 6.
c) P contains a cycle of length 7.
d) The connectivity ofP equals the edge-connectivity ofP.
3. a) Any line graph is claw-free.
b) In every network, the maximum value of a feasible flow equals the minimum capacity of a source/sink cut.
c) Any loopless bipartite graphGhas a bipartite subgraph with at least a half of the edges ofG.
d) There are only five Platonic solids (a polyhedron in dimension 3 such that all faces are the same regular convex polygon and that all vertices have the same degree): tetrahedra, cube, octahedra, dodecahedra, and icosahedra.
4. a) The degree of every vertex in any 6891-regular graph must be even.
b) The chromatic polynomial of any graph of order 2019 has no real root larger than 2018.
c) No simple graph of order 2019 with a million edges is triangle-free.
d) The number of trees with vertex set {1,2,3,4,5}, up to isomorphism, is more than 200.
5. a) The graph theorist Gabriel Andrew Dirac lived in an early age of the mathematician Leonhard Euler.
b) Our class took place in more than 4 classrooms, excluding where you are sitting in. c) Our teaching assistant solved every problem quite smoothly on blackboard in class. d) In doing problem sets as homework, graphing by using Tikz is more recommended
by the instructor than that by inserting a photo.
IV (30 points) Letnandkbe integers such thatn≥11 and 1≤k < n/2. Denote by
Zn={0,1, . . . , n−1}
the additive group of integers modulon. A generalized Petersen graph, denotedP(n, k), is the undirected graph with vertex set
V ={ui:i∈Zn} ∪ {vi:i∈Zn}
and edge setE=O∪I∪S, where
O={uiui+1:i∈Zn}, I={vivi+k:i∈Zn}, and S={uivi:i∈Zn}.
1) (20 points) Find the distance between the verticesu0 andu5 inP(11,3), and the girth of
P(12,4).
2) (7 points) Letx, y∈V. Letpbe a shortest path fromxtoy. Prove or disprove thatp
contains at most two edges inS.
3) (3 points) Letap be the number of arcs inpof the formuiui+1 orvivi+k, and bp be the
number of arcs of the formui+1ui orvi+kvi. Prove or disprove that min(ap, bp)≤k/2.
Bonus
The lazy Tuatua problem (An extra 100 points). Duadua and Tuatua the puppies play on thebitplayground. David draws 5 circles on the ground such that any two circles intersect at two distinct points, and no three circles pass through the same point. Duadua runs in the following fashion:
• Duadua chose a non-intersecting point on a circle to start, clockwise.
• He keeps running along the current circle until an intersecting point is reached.
• Duadua continues his rushing on the new circle, and also changes from clockwise to counterclockwise or vice versa.
The particular drawing of David enables energetic Duadua to cover all the circles. Now, lazy Tuatua wish to wander on and cover 4 of the circles in the same fashion, and never to walk along the remaining circle. Does her wish possibly come true? What if Duadua fails?
Scoring rules and comments:
• The full score of this examination is 100 points.
• Section I: It would be quite uncommon that someone obtains a recommendation letter from the instructor but fails to obtain a full score from this section.
• Section II: Correctly stating any one definition/theorem gives you 20 points, and another one 10 points. Extra statements would not yield extra points. It would be uncommon if you are satisfied with your points in the case that your wording was considered to be vague or to cause ambiguity.
• Section III: Letnbe the number of problems in the first four that you correctly solve. The points you obtain from this section will be
(
min(3 + 11n−n2, 30), if your answer to the last problem is correct;
n(19−n)/2, otherwise.
• Section IV: A pure guessing for any answer would not produce a point.
• The bonus: Any major progress or resolution of the lazy Tuatua problem could result in extra points at the instructor’s discretion.
Afterthoughts: The score distribution looks Guassian from some point of view. Among 9 master or doctoral students, the highest score was 88 points. The instructor summarized bad and good parts of this collection of problems as the final examination as follows.
Bad part:
• The amount of problems is too much to finish in 60 minutes. The real examination run 70 minutes. The lazy Tuatua problem, nevertheless, received no attention in the 70 minutes.
• Problem IV 1) might be too skillful.
Good part:
• The selected examination time length of 60 minutes was enough to provide a reasonable point.
• Unpopular but popular are concept and theorem statements and choice problems.
• Many others, as students said.
