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Unformatted text preview: 2. if G = (V, E) is a connected graph with E = i? and
degto} 3 3 for all a E V, what is the maximum value for W l? 10. For a E 2+, how many distinct (though isomorphic) paths
of length 2 are there in the ndirnensional hypercube Q”? 12. a] Porn 3 2. let V denote the vertices in Q". For 1 5 I: e
t 5 at deﬁne the relation ER on V as follows: If to. :r E 1’.
then to ER x if at and x have the same bit [0. or 1] in position
it and the same bit [0, or i] in position if of their binary la
bels.[Forexan1ple.ifn = T andk = 3.3 = tithen “00010
0% 0000011.] Show that 9% is an equivalence relation. How
many blocks are there for this equivalence relation“? How
many vertices are there in each block? Describe the sub—
graph of Q" induced by the vertices in each block. 18. Let i: be a ﬁxed positive integer and let G = (V. E} be
a loopfree undirected graph, where degtv] 2; k for all a E V.
Prove that G contains a path of length it. 6. Let a E 2* with n a 4. Hovtr man}r subgraphs of K" are
isomorphic to the complete bipartite graph K 1.3? 18. Let G = {V1 E} be an undirected connected loopfree
graph. Sappose further that G is planar and determines 53 re— gions. If, for some planar embedding of (I. each region has at
least ﬁve edges in its boundary, prove that V a 32. 21. Prove that every loopvfree connected planar graph has a
vertex v with deg(a) e n. 28. Let G = (V, E } be a loop—free connected planar graph. If
U is isomorphic to its dual and V = n. what is IEI‘EI ...
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This note was uploaded on 10/01/2011 for the course MACM 101 taught by Professor Pearce during the Spring '08 term at Simon Fraser.
 Spring '08
 PEARCE

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