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# Doc7 - 2 if G =(V E is a connected graph with |E| = i and...

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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 n-dirnensional 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 loop-free 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 loop-free 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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Doc7 - 2 if G =(V E is a connected graph with |E| = i and...

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