Andrew Thomason

Frank Ramsey

The name of Frank Ramsey is universally known amongst combinatorial mathematicians, but our casual mental picture of him can easily be an unimpressive one the man who almost stumbled across the theorem that now bears his name, thereby anticipating Erdos and Szekeres, who of course gave the proper proof. Such an idea of Ramsey is entirely false: he was an absolutely brilliant man, who would certainly have become even more famous had he not died so young, and who would surely, it could easily be argued, have made yet further remarkable contributions to philosophy, economics and logic and to combinatorics.

Weakly Pancyclic Graphs

A graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. A substantial result of Haggkvist, Faudree, and Schelp (1981) states that a Hamiltonian non-bipartite graph of order n and size at least ?(n?1)2/4?+2 contains cycles of every length l, 3?l?n. From this, Brandt (1997) deduced that every non-bipartite graph of the stated order and size is weakly pancyclic. He conjectured the much stronger assertion that it suffices to demand that the size be at least ?n2/4??n+5. We almost prove this conjecture by establishing that every graph of order n and size at least ?n2/4??n+59 is weakly pancyclic or bipartite.

On the girth of Hamiltonian weakly pancyclic graphs

A graph is called weakly pancyclic if it contains cycles of all lengths between its girth and circumference. In answer to a question of Erdos, we show that a Hamiltonian weakly-pancyclic graph of order n can have girth as large as about . In contrast to this, we show that the existence of a cycle of length at most - 1 is already implied by the existence of just two long cycles, of lengths n and n - 1. Moreover we show that any graph, Hamiltonian or otherwise, which has n + c edges will have girth of order at most (n/c)log c.

A Remark on the Number of Complete and Empty Subgraphs

Let kp(G) denote the number of complete subgraphs of order p in the graph G. Bollobas proved that any real linear combination of the form ∑apkp(G) attains its maximum on a complete multipartite graph. We show that the same is true for a linear combination of the form ∑apkp(G) +bpkp(G¯), provided bp≥0 for every p.