Arrived home in Singapore from Appleton, Wisconsin this morning at about 7:15am. I'm now dead tired. Will try to catch some sleep soon. Tomorrow is a big day .. eve of Chinese New Year! Looking forward to the reunion dinner.
Was craving for buffalo wings so went to Hooters (1272 North Casaloma Drive, Appleton, Wisconsin) for dinner tonight. I know what you're thinking, but they do have ok food. I especially enjoyed the Carolina style oyster roast. The oysters were rubbed with salt and then roasted in their shells, and served with drawn butter, lemon, orange, or cocktail sauce. It was great fun shucking them! Some of the oysters were a little dry but the juicy ones were wonderful. The servings were reasonable. We paid about US$7.50 for about 15-18 oysters. The buffalo wings were not bad but I had too much of them and am suffering from some wings overdose right now.
With respect to the problem described, Jan Haugland and I have been corresponding a little bit. We managed to show that 2/(f(d)+2) <= f(d+1)/f(d) < 1 for all d. We think lim f(d+1)/f(d) exists. This follows if lim f(d) = 0, which we presume would not be too difficult to resolve. Jan thinks f(d) grows roughly like Alog(d)/d.
Considering planar graphs embedded on a sphere, we see that g(d+2) <= 2e by going through all the faces. This inequality gives f(d) >= 2/(d+2). The following construction gives a better bound on f(d).
Given d, let n be the smallest prime power such that d <= n^3-1. Now take a projective plane of order n and consider its point-block incidence graph G. G has v = 2(n^2+n+1) vertices, e = (n+1)(n^2+n+1) edges, and girth 6. Now, e-v = n^3-1 >= d. Since f is a decreasing function, we now have f(d) > 6/((n+1)(n^2+n+1)) = (6/d)*(1-2/(n+1)) = (6/d)*(1-2/(d^(1/3)+1)).
Went to Trim B's (201 S. Walnut St, Appleton, Wisconsin) for fish fry tonight. Food was average. This is an over-rated restaurant, frequented by mainly older people.
Was at El Azteca (840 Fox Point Plaza, Neenah, Wisconsin) for dinner tonight. Good authentic Mexican food at very reasonable prices. Decoration is colorful and waiters are friendly. If you are celebrating your birthday, let them know and you will be in for a surprise! The pina colada, made with real coconut milk and crushed pineapples, is out of this world! During happy hours from Sun to Thur, 5pm-7pm, magaritas go for US$3 each. This place is packed on weekends.
Tasted a bottle of Bogle Petit Sirah, California 2000 tonight. Bought for US$8.99 at World Market, Appleton, Wisconsin. Ruby red in color. Spice, grass & berries on the nose. Medium body and peppery. The tannin is not strong enough to balance the strong acidity.
On 17 Jan 2003, Jan Kristian Haugland posted the following interesting problem on sci.math.research.
For a graph G, let d(G) = |E(G)|-|V(G)| and define
He observed that:
i) f(d) = 0 for d < 0;
ii) f(0) = 1;
iii) f(1) = 2/3;
iv) f(2) = 1/2;
and asked if f(3) = 4/9 (as achieved by K_3,3) and f(5) = 1/3 (as achieved by the Petersen graph).
I can show that 2f(d)/3 <= f(d+1) <= f(d). When I have time, I will investigate this problem further.