### On the Number of Complete Subgraphs and Circuits in a Graph

D. G. Larman
1969 Proceedings of the Royal Society A
E rd o s ( 1964) co n je c tu re d t h a t a g ra p h w ith nk n odes each o f v alence a t least (nl)fc co n ta in s k d isjo in t com plete su b g rap h s each w ith n nodes. T h is a n d a re la te d c o n jec tu re of G riinbaum are discussed a n d p ro v ed in som e special cases. Som e sim ilar resu lts are o b ta in e d show ing th e existence o f d isjo in t c ircu its in g ra p h s w ith sufficiently hig h valences. I n t r o d u c t i o n Erdos (1964) (see problem 9) made the
more » ... ) made the following conjecture: for given natural numbers n, k, the number k is the largest integer m for which the following proposition holds: I f G is a graph with having valence of at least nk -m, then G contains k disjoint compl graphs, each having n nodes. Throughout this paper we shall only be considering finite undirected graphs without loops or multiple edges. For n = 2 the conject Dirac (1952) and for n -3, it was established by Corradi & H ajnal (1 baum (1968) considers the dual colouring problem and makes what at first, appears to be a more general conjecture: Every graph G satisfying the conditions (a) it has nk + r nodes, 0 ^ r ^ k, (b) no node has valence ^ m, can be k-coloured, each colour appearing at least n and at most 1 times. In fact it is not difficult to show (Theorem 1 (i)) th a t if Erdos's conjecture is true then so also is Griinbaum's conjecture. He proved this conjecture for ^ 4 and raised the problem as to whether or not even the cases k = \_\ri\, k = [^w] can be deduced from the results of Dirac and Corradi-Hajnal. This paper will be concerned with proving partial results for these conjectures and will go on to consider the more general problem of the existence of circuits in graphs. I t will be convenient to have the following reformulation of Grunbaum's conjecture: k is the largest integer m for which the following proposition holds :IfG is a graph with nk + r nodes, 0 < r < k, each node having valence les than m, then G can be k-coloured, each colour appearing at least n and at most n+ 1 times. f R e s e a rc h s u p p o r te d b y a H a rk n e s s F e llo w s h ip o f th e C o m m o n w e a lth F u n d . [ 3 2 7 ]