Graph Theory Problem Ideas

Transcription

1 Graph Theory Problem Ideas April 15, 017 Note: Please let me know if you have a problem that you would like me to add to the list! 1 Classification Given a degree sequence d 1,...,d n, let N d1,...,d n denote the number of isomorphism classes of graphs with that degree sequence. Find a formula for N d1,...,d n. Find an algorithm that computes a graph in each isomorphism class. Example. In the case of the degree sequence,,,,,,,,, the four di erent isomorphism classes are drawn at the top of page 65. We could write this as N 9 =4. Similarly,if you solve Exercise 3.16, you should realize that N 6 9 = 4 (hint: use Theorem 3.1). What is the symmetry at work here? Can you generalize this to get an identity for N d1,...,d n? Remarks. Start with simple degree sequences! For example, compute N 1 n and N n. (r-regular graphs with n vertices) would already be ex- Solving this problem for N r n tremely interesting. Ideally, you would find a closed formula for N d1,...,d n in terms of the variables d 1,...,d n. If a closed formula proves to be elusive, it may be possible to show that the numbers N d1,...,d n for various choices of degree sequences can be assembled into a nice structure, such as a power series that has a nice factorization. One approach to computing N d1,...,d n is to find relationships between these numbers as you vary the degree sequences. For instance, you could investigate how the number changes when you subtract 1 from d 1. Sometimes small changes to a counting problem can lead to a much nicer formula. For instance, counting all pseudographs with a given degree sequence or counting only connected graphs with a given degree sequence may be easier. Checking isomorphism Devise an e cient algorithm to determine whether two graphs are isomorphic.

2 Remarks. The brute-force algorithm we saw in class takes n!time,whichisconsidered to be extremely bad. You should aim for a polynomial-time algorithm (e.g. n 3 ). Once again, restricting to a special class of graphs (such as r-regular graphs or connected graphs) may make this problem more tractable. 3 Graphical degree sequences Devise a list of conditions for degree sequences such that every degree sequence satisfying those conditions is graphical. Remark. We already have an e cient (linear time) algorithm for determining whether a sequence is graphical. You could try to use this algorithm to help you devise the list of conditions. 4 Properties of a graph and its adjacency matrix How are properties of a graph (such as the degree sequence, connectedness, bipartiteness, etc.) reflected in the adjacency matrix? Conversely, how are properties of the adjacency matrix (such as invertibility, eigenvalues, eigenvectors, etc.) reflected in the associated graph? 5 Graceful trees Does every tree have a graceful labeling? (For the definition of graceful labeling, see Section 8.3.) 6 Partitioning edge-connectivity Let G =(V,E) be an (a + b +)-edge-connected graph. Does there exist a partition of E into sets A, B so that the subgraph (V,A) is a-edge-connected and the subgraph (V,B) is b-edge-connected? (This problem was submitted to the Open Problem Garden by Matt Devos.) Example. K 5 is (1+1+)-edge-connected. Letting A be the edges in any 5-cycle in K 5, the remaining edges of K 5 also form a 5-cycle. Cycles are 1-edge-connected (which just means connected), so the partition in the question does exist for K 5. 7 Decomposing an Eulerian graph into cycles Is it true that every (simple) Eulerian graph of order n can be decomposed into apple n 1 cycles? (In the Open Problem Garden, this problem is attributed to György Hajós.) Example. Acompletegraphofoddorder(n =k +1) can be decomposed into exactly k = n 1 cycles.

3 8 Hamiltonian cycles Devise a polynomial time algorithm for determining whether a graph has a Hamiltonian cycle. Remark. This is an extremely hard problem in general. As usual, it could be helpful to restrict to an interesting class of graphs instead of general graphs. 9 Ramsey numbers Develop an e cient algorithm to compute Ramsey numbers (such as r(k 3,K 10 ), r(k 4,K 6 ), or r(k 5,K 5 )). Alternatively, invent a formula (such as Theorem 11.8 in the textbook) for a certain class of Ramsey numbers and prove your formula is correct. Remark. Warning: this problem is harder than it looks! You might think that r(k 5,K 5 ) cannot be that hard to compute using a computer because K 5 is a fairly small graph, but you should realize that this Ramsey number is at least 43 and that there are (43 ) possible red-blue colorings of K 43.Onethingyoucouldtrytousetoyouradvantageisthat K 43 has a lot of symmetry; in fact its automorphism group consists of all permutations of 43 vertices, which is a group of order 43! Unfortunately, this is much much much smaller than the number of colorings, so even a brute force approach that makes use of this symmetry has no chance of considering all possible colorings. 10 Domination numbers Develop an e cient algorithm for computing the domination number of a graph. Alternatively, try to prove one of the many conjectural inequalities such as the following: does every 3-connected cubic graph have domination number at most dn/3e, where n is the order of the graph? 11 Game theory Choose a game such as Tic-tac-toe, Connect Four, Chinese checkers, Checkers, and Chess. Construct a graph whose vertices correspond to positions in the game. Two game positions are connected by an edge if there is a legal move in the game that takes one position to the other. Once the graph is constructed, determine optimal play by working backward from the winning and drawing states. Studying the isomorphisms of this graph induced by symmetries of the game could also be interesting. Remark. You should probably start with a simple game like Tic-tac-toe before moving to more complicated games whose graphs of positions are larger. You probably want to use a computer program to store the game positions, construct the graph, and analyze the graph. 3

4 1 Large-scale graph theory One trendy research area is studying large graphs arising in the real world (for instance networks like the internet and Facebook or models of the neurons in a brain). Depending on the particular application, there are di erent graph-theoretic questions to pursue. For instance, which Facebook users are most influential in the spread of trends and memes? Can the neurons in the brain be grouped in clusters corresponding to di erent brain functions? Example (Google s PageRank algorithm). There is an excellent article on the algorithm that Google uses to rank search results ( kogan/teaching/m430/ GooglePageRank.pdf). The basic idea is to construct the directed graph whose vertices are webpages and direct edges are links between webpages. The challenge is to develop a way to rank the importance of webpages so that a search algorithm can decide which results should be displayed first. The importance of a webpage is based on the number of links leading to that page, weighted by the importance of each of the pages from which the link is coming. But how can we use the importance of webpages in the definition of importance? It turns out that there is a clever mathematical way to make sense of this: one constructs a certain matrix from the data of the graph and computes a certain eigenvector of that matrix. The entries of that eigenvector, which turn out to all be positive, give the desired ranking. The key theorem in linear algebra that makes this work is the Perron-Frobenius theorem, which you can read about in Wikipedia. 13 Spectral graph theory Expanding on Problem 4, how do the properties of other matrices associated to a graph reflect the properties of the graph? Remark. The word spectral refers to the eigenvalues of these matrices, which often encode interesting information about the graph. Example. According to the Matrix-Tree Theorem, the number of spanning trees in a graph is the product of the nonzero eigenvalues of its Laplacian matrix. 14 Coverings of graphs In topology, a covering of a graph G is a continuous map H! G, where H is a graph, that sends vertices to vertices and edges to edges such that each vertex and edge of G has the same number of preimages in H. These coverings correspond to subgroups of the fundamental group 1 (G) of G, which parametrizes distinct loops in G. There are three fields of math at play here: graph theory, topology, and group theory. The goal of this problem is to study how the correspondence between coverings and subgroups works for various graphs. How do properties of the graph, the cover, and the subgroups relate? 4

5 15 Kempe chain action Study the action of Kempe chain swaps on the set C(G, k) of all k-colorings of a graph G. One question to think about: for which groups G and which values k is the action transitive (transitive means that for any starting coloring C and given any coloring C 0, there is a sequence of Kempe chain swaps that takes C to C 0 )? Suppose G is given a k-coloring C. Giventwocolorsi and j and a vertex v V (G), the i-j Kempe chain H i,j (v) containingv is the induced subgraph of G containing all vertices w such that there is a walk from v to w and each vertex along the walk has color i or j. The special property of H i,j (v) isthatinterchangingthecolorsi and j of its vertices produces another valid coloring C 0 of G. We can say that the action of swapping colors along the Kempe chain H i,j (v) changesthecoloringfromc to C 0. Remark. The permutation group also acts on C(G, k) bypermutingcolors. Onecanshow that the action of a permutation on a coloring C can be expressed in terms of Kempe chain swaps. This is because given a coloring C, swapping along all i-j Kempe chains has the e ect of interchanging i and j in the full coloring, and the transpositions of colors generate the permutation group. 5