(1) 15* Remarks on a Paper of Pósa

This note will use the terminology of PÓSA'S paper. Gin) will denote a graph of n vertices and l edges and G(,n)(k) denotes a graph of having v vertices 1 edges and every vertex of which has valency > k. ORE [2] proved that if l > (n-1 f + 2 then every Gin) is Hamiltonian, and he showed that n-1 the result is false for l = I Now I prove the following more general Theorem. Let 1 < k < n/2. Put j 1L-t h=1-{-max 1 2 +1-'I= her< 2 n-1 n.-k 1 n-1 1 max 1 k2.-i-] Then every Gjk ~(k) is Hamiltonian. There further exists a Grk?,(k) which, is not Hamiltonian. First of all observe that by the theorem of DIRAC (see the preceding paper of PÓSA) if every vertex of G has valency > n/2 then a is Hamiltonian, thus the condition 1 < k < nl? can be assumed without loss of generality .-Next a simple computation shows that nn t I t 2 decreases for l < t < (n-2)j3 and increases for (u-2)/3 < t < n,/2, which proves the second equality of (1). Now eve are ready to prove our Theorem. If our G (") (k) is not Hamiltonian then by the Theorem of PÓSA there exists a t, k < t < nj2 so that G00(k) has at least t vertices x,,. .. . x 1 of valency not exceeding t. The number of edges of G0 ")(k) which are not incident to any of the vertices x l BLOCKINx, is clearly at most Tt f) (i. e. if the vertices of G (") (k) are x1 BLOCKINx" we obtain n ,, tl edges if every two of the vertices. r i and x1. ., t < j 1 < j2 <-n are connected 22,