June 22, 2008
Sometime back I read a short paper by Chvatal and Erdos that presents a couple of sufficiency conditions for hamiltonicity.
One of the results there is – If G is an s-connected graph with at least 3 vertices, and has a maximum independence number of s, then G has a hamiltonian circuit.
It is an interesting result connecting robustness and hamiltonicity. A graph is s-connected if there are s edge independent paths between any two nodes in the graph. In other words, it is the size of the minimum edge cut i.e. the minimum number of edges whose deletion increases the number of components in the graph.
The maximum independence number of a graph is the size of the biggest independent set. An independent set is a set of vertices of the graph such that no two vertices in the set are adjacent. In other words, there is no edge between any pair of vertices in the independent set. The bigger the independence number, the easier it is to fragment the network.
As you can see both these concepts can be used to measure the robustness of a graph. Also, it is not unnatural that connectivity and hamiltonicity are related. Intuitively, the more independent paths, the greater the chances of a graph being hamiltonian. And, a hamiltonian circuit is at least 2-connected: a circle is the simplest hamiltonian circuit and it is 2-connected.
Well, but the point of this post is something else really. The paper I referred to in the beginning is just a 3-page note. And it has an interesting footnote on the 1st page. The footnote says: This note was written in Professor Richard K. Guy’s car on the way from Pullman to Spokane, Wash. The authors wish to express their gratitude to Mrs. Guy for smooth driving.
March 17, 2008
Alright, the title is meant to be catchy; there is no real connection there. Except that it is in the context of paying my apartment gas bill I found out about a simple application of modulo arithmetic in electronic transactions in the US. (I am not talking about cryptography. Nowhere near that.) While trying to pay the gas bill through an e-cheque, I found that I had to fill in something called a “routing number” before submitting the e-cheque. Having never heard of the term before, I searched a bit and found out that the number serves chiefly as an identifier of a bank (or other financial institutions) in electronic transactions. It is a 9 digit number which identifies the location of a bank and the bank. (I think a bank can have multiple routing numbers.)
The modulo arithmetic part comes in in generating and validating a routing number. To validate a routing number the following checksum is used:
[3(d1 + d4 + d7) + 7(d2 + d5 + d8) + 1(d3 + d6 + d9)] mod 10 = 0
I don’t know why this particular checksum. And I think, the the first 8 digits are known and the last digit is generated using the above formula. Not sure who does this.
[Source: wikipedia and official ABA policy (pdf link)]
Incidentally, Gauss was responsible for modulo arithmetic (or clock arithmetic). The book The Music of the Primes has a nice description of this and many other beautiful things that Gauss developed. On this note, I also recall a quote in the brilliant book called Concrete Mathematics. In that book, the authors have included what are called “marginal notes” — they are actually the notes/comments written by the first set of students who got to read the draft of the book. The quote is – “It seems a lot of stuff is attributed to Gauss; either he was really smart or he had a great press agent.” On the same page, there is another marginal comment – “Maybe he just had a magnetic personality.”
March 11, 2008
“To be conscious that you are ignorant is a great step to knowledge.”
“There is much pleasure to be gained from useless knowledge.”
“When you know a thing, to hold that you know it; and when you do not know a thing, to allow that you do not know it – this is knowledge.”
“The beginning of knowledge is the discovery of something we do not understand.”
“One’s first step in wisdom is to question everything – and one’s last is to come to terms with everything.”