By M.Rosenfeld, J.Zaks
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Extra info for Convexity and graph theory: proceedings of the Conference on Convexity and Graph Theory, Israel, March 1981
Math. 19 (1974) 292-296. Received 25 April 1Y8J Annals of Discrete Mathematics 20 (1984) 37-41 North-Holland SOME CONDITIONS FOR DIGRAPHS TO BE HAMILTONIAN D. AMAR, I. FOURNIER and A. GERMA Laboratoire de Recherche en Znformatique, Universite' de Paris-Sud, 91405 Orsay Cedex, France We answer a question of B. Bollobas and we determine the minimum number of arcs in a diagraph with given minimum indegree and outdegree guaranteeing this digraph to be Hamiltonian or strongly Hamilton-connected. 1. Introduction The notation we use can be found in [ l ] In .
X, be the vertices of G" and suppose that u, u and w are the degrees of xl, x2 and x3, respectively. 3,there are at most two vertices xi with i 3 4 that are adjacent to xl, x2 and x3. Thus, the number of edges that join xl, x2 and x j to some x,, i 2 4 , is at most 2 . 3 + ( n - 5 ) . 2 = 2 n - 4 , and we obtain: 0 u+u+w<6+(2n-4)=2n+2. Proof of Theorem 1. Note that if d l , d2,.. ,d, are the degrees of the vertices of a graph G", then (3, N(G",K1,k)=$ i=l for all k 2 2 . Therefore f ( n , K , , k )is just where the maximum is taken over all degree sequences of planar graphs on n vertices.
Lesniak-Foster, Graphs and Digraphs (Prindle, Weber & Schmidt, Reading, 1979).  F. Buckley, Self-centered graphs with a given radius, Proc. 10th S-E Conf. Combinatorics, Graph Theory, and Computing, Congressus Numerantum XXIII (Utilitas Mathematica. Winnipeg, 1979) pp. 211-215.  F. Harary, Graph Theory (Addison-Wesley, Reading, 1969).  J. Mycielski, Sur le coloriage des graphes, Coll. Math. 3 (1955) 161-162. Received March 1981 This Page Intentionally Left Blank Annals of Discrete Mathematics 20 (1984) 25-36 North-Holland ON THE NUMBER OF SUBGRAPHS OF PRESCRIBED TYPE OF PLANAR GRAPHS WITH A GIVEN NUMBER OF VERTICES N.