Unformatted text preview: Qualifying Exam, Fall 2005
Graph Theory and Combinatorics 1. (a) Deﬁne a (b, v,r,k,)\)—design.
(b) Derive the following identities:
I l bk = or r(k— 1) = /\(v— 1) (c) What are the symmetric designs called for which k = 3 and /\ = 1? Derive necessary congruence
conditions on v for such a design to exist. 2 (a) Prove Euler’s formula relating the numbers of vertices, edges and faces of a connected planar
map. (Hint: induce on the number of edges.) (b) From this, derive the formula 1 1 1 1
+=+—,
r s e 2 where r denotes the average valence, 3 denotes the average covalence, and 6 denotes the number of
edges.
(c)1Prove that every ﬁnite planar map has a face of covalence at most 5.
(d) Prove that if a planar map has no face with covalence less than 5, then it has at least 12 faces
of covalence exactly 5. Finally, conclude th this bound is sharp.
”I no \Sol VLP‘l" L25
3. (a) Prove that if a graph‘ifs edge transitive but not vertex— transitive, then it is bipartite.
(b) Prove that Cayley graphs are vertex— transitive but not necessarily edgetransitive.
(c) Prove that connected, edgetransitive graphs have trivial atoms. 4. .Let X be a connected k valent graph with diameter 6, where 19,6 > 2. (a) As a function of k and 6, what is the largest possible value of lV(X )? (b) What is the girth of such a maximal graph? . (c ) For what values of k and 6 is this bound known to be attainable, and for what values is it known
not to be attainable? (d) Is such a graph distance—regular, and if so, what is its distance array? .. 5. Let positive integers n, m and k be given with k < 71, let A1, . . . ,Am be disjoint nsets and let
E = UlilAi Deﬁne I Q E to belong to I if }I ﬂAlI S k for all 2'. (a) Prove that I is the collection of independent sets for a matroid on E. (b) Describe the basfs and ran> function for this matroid and the cycles of its dual. 6. How many integer solutions are there to the equation $1+$2+$3+$4+$5=20 {3 where 1 3 mi 3 6, for 2' = 1, 2,3,4? [You need not execute all of the arithmetic. It sufﬁces to
indicate what calculations need to be performed] ...
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 Combinatorics

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