Summary: The Longest Cycle of a
Graph with a Large
MATHEMATICS RESEARCH CENTER, AT&T
BELL LABORATORIES,MURRAY HILL, NJ 07974
We show that every graph G on n vertices with minimal degree at
least n/k contains a cycle of length at least [n/(k - 111. This verifies
a conjecture of Katchalski. When k = 2 our result reduces to the
classical theorem of Dirac that asserts that if all degrees are at least
in then G is Hamiltonian.
For a graph G = (V(G),E(G))let 6(G)denote the minimal degree of vertex of
G and let c(G) denote the circumference, i.e., the size of a longest cycle of G.
In this note we prove
Theorem 1.1. Suppose n > k 2 2 are integers and that G is a graph on n ver-
tices with 6(G) 2 n/k. Then c(G) 2 [n/(k - l)].
This result was conjectured by M. Katchalski , who also proved it for
k I4.For k = 2 it reduces to the classical theorem of Dirac (,-see also