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....(1 o(1) In Section 4 we include a sketch of the proof of Theorem 4. Note that for k = 2, 3 and 5, Theorem 4 is, apart from the factor c k in the exponent, best possible because, for these values of k, there are fC 4 ; C 2k g free graphs with Theta(n edges (see Benson [3] and Wenger [22]) and 2 ex(n;fC4 ; C 2kg) is an obvious lower bound for jForb n (C 4 ; C 2k )j. For general k we remark that a conjecture of Erdos and Simonovits [10] states that ex(n; fC 4 ; C 2k g) Omega Gamma n ) It is crucial for the proof of Theorem 4 that the graphs under ....

R. Wenger. Extremal graphs with no C

....7 (Theorem 7.3) we show that weighted graphs also have 3 spanners of size O(n 3=2 ) We present there an algorithm, whose running time is O(mn 1=2 ) for finding such 3 spanners. As there are bipartite graphs with Omega Gamma n 3=2 ) edges that do not contain cycles of length four [Wen91] this result is tight, up to polylogarithmic factors. We can also show: Theorem 6.3 Every unweighted undirected graph G = V; E) on n vertices has a 4 emulator with O(n 4=3 ) edges. Such a graph can be constructed in O(n 7=3 ) time. Proof: It is not difficult to check that the graph ....

....1 ; w 2 ) 2) ffi G (w 2 ; w 3 ) 2) ffi G (w 3 ; v) 1) ffi G (u; v) 6 : This completes the proof of the theorem. 2 It is easy to see that k emulators are Steiner (k 1) spanners. It follows easily from the arguments of Althofer et al. ADD 93] and the constructions of Wenger [Wen91] that there are unweighted undirected graphs on n vertices for which every Steiner 3 spanner, and therefore any 2 emulator, must have Omega (n 3=2 ) edges, and there are graphs for which every Steiner 5 spanner, and therefore any 4 emulator, must have Omega Gamma n 4=3 ) edges (where ....

R. Wenger. Extremal graphs with no C

....number of edges of an n vertex graph of girth at least 2t 1. The upper bound of Bondy and Simonovits [4] states ex(n; C 2t ) O(n 1 1 t ) This is conjectured to give the correct order of magnitude. However, this is supported by a construction only for t = 2; 3 and 5 (see, e.g. in Wenger [45], or TUR AN PROBLEMS FOR WEIGHTED GRAPHS 3 Bollob as [3] Erdos gave a probabilistic lower bound ex(n; C 2t ) Omega Gamma n 1 1=2t ) In the remaining cases the best lower bound is given by the so called Ramanujan graphs of Lubotzky, Phillips and Sarnak [35] Margulis [37] and Imrich ....

R. Wenger, Extremal graphs with no C

....: In Section 4 we include a sketch of the proof of Theorem 4. Note that for k = 2, 3 and 5, Theorem 4 is, apart from the factor c k in the exponent, best possible because, for these values of k, there are fC 4 ; C 2k g free graphs with (n 1 1=k ) edges (see Benson [3] and Wenger [22]) and 2 ex(n;fC4 ; C 2k g) is an obvious lower bound for jForb n (C 4 ; C 2k )j. For general k we remark that a conjecture of Erd os and Simonovits [10] states that ex(n; fC 4 ; C 2k g) n 1 1=k ) It is crucial for the proof of Theorem 4 that the graphs under ....

R. Wenger. Extremal graphs with no C

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R. Wenger. Extremal graphs with no C

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R. Wenger. Extremal graphs with no C

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R. Wenger. Extremal graphs with no C

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R. Wenger. Extremal graphs with no C

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