, Volume 1, Issue 3, pp 156-160

On the structure of linear graphs

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Denote byG(n; m) a graph ofn vertices andm edges. We prove that everyG(n; [n 2/4]+1) contains a circuit ofl edges for every 3 ≦l<c 2 n, also that everyG(n; [n 2/4]+1) contains ak e(u n, un) withu n=[c 1 logn] (for the definition ofk e(u n, un) see the introduction). Finally fort>t 0 everyG(n; [tn 3/2]) contains a circuit of 2l edges for 2≦l<c 3 t 2.

This work was done while the author received support from the National Science Foundation, N.S.F. G.88.