Search Results

Eulerian graphs with $e$ edges cannot be cordial unless $e$ is a multiple of $4$ so don't bother looking at $4$-regular graphs with an odd number of vertices. (This is in Cahit's original paper that i …

At the moment I am interested in maximal non-hamiltonian graphs, so that is a (simple, undirected) graph that does not itself have a hamilton cycle, but if you add an edge between any two distinct non …

My computer tells me that there 195291625 graphs on 20 vertices with minimum degree at least 3 and maximum degree at most 6 and girth at least 5. Sadly none of them have chromatic number 4, I just ge …

They are called quartic Möbius ladders. They are one of the fundamental classes in Johnson & Thomas's classification of internally 4-connected graphs, and crop up in matroid theory for the same reaso …

Many years ago, Grinberg found some uniquely-hamiltonian $3$-connected graphs, and published his results in a paper that has been cited several times as follows. E. Grinberg, Three-connected graph …

I have now resolved most of the mysteries, and as MO prompts me to answer my own question, I am now doing so even though it feels a bit odd. After some false starts with expired email addresses, I ma …

OK, I can now confirm that even your modified conjecture is false, although the first counterexample has $13$ vertices. First we need to know which complete $k$-partite graphs on $13$ vertices are cop …

I'm not sure why such an old question suddenly bubbled up to the (my) first page after several years, but I won't let that stop me answering it. As explained above, Rota's critical problem is usually …

I tried to go through Birkhoff and Lewis many years ago but it is not easy because they use different variables and the style is so different to modern proofs. In modern terms, the key idea is that i …

This graph is known as the half-cube. I don't know about the other question.

22 vertices, there are 80 of them. Jensen and Royle, Small graphs with chromatic number 5 : a computer search Journal of Graph Theory, 1995.

In a short 1946 paper "On Hamiltonian Circuits", Tutte proved the famous result that an edge in a cubic graph lies in an even number of Hamilton circuits. He attributed the result to his friend CAB S …

You can compute these numbers for small $n$ and $r$ using Pólya's Enumeration Theorem (which is just a specific application of the orbit-stabiliser theorem). A simple binary matroid of rank at most $r …

This is a question that seems like it should have been studied before, but for some reason I cannot find much at all about it, and so I am asking for any pointers / references etc. The determinant of …

Yes, this is true, but I don't know a reference, so here's a proof (I think). Let $$ R(A, x) = \frac{x^T A x}{x^T x} $$ be the Rayleigh quotient. We know that for a symmetric (in fact Hermitian) matr …