# On the structure of linear graphs

## Authors

- Received:

DOI: 10.1007/BF02759702

- Cite this article as:
- Erdös, P. Israel J. Math. (1963) 1: 156. doi:10.1007/BF02759702

## Abstract

Denote by*G*(*n; m*) a graph of*n* vertices and*m* edges. We prove that every*G*(*n*; [*n*
^{2}/4]+1) contains a circuit of*l* edges for every 3 ≦*l*<*c*
_{2}
*n*, also that every*G*(*n*; [*n*
^{2}/4]+1) contains a*k*
_{e}(*u*
_{n}, u_{n}) with*u*
_{n}=[*c*
_{1} log*n*] (for the definition of*k*
_{e}(*u*
_{n}, u_{n}) see the introduction). Finally for*t*>*t*
_{0} every*G*(*n*; [*tn*
^{3/2}]) contains a circuit of 2*l* edges for 2≦*l*<*c*
_{3}
*t*
^{2}.