Lower bounds for the clique and the chromatic numbers of a graph

G is any simple graph with m edges and n veriices where these vertices have vertex degrees d(l)zd(2)r...rd(n), respectively. General algorithms for the exact calculation of x(G), the chromatic number of G, are almost always of ‘branch and bound’ type; this type of algorithm requires an easily constructed lower bound for x(G). In this paper it is shown that, for a number of such bounds (many of which are described here for the first time), the bound does not exceed cl(G) where cl(G) is the clique number of G. In 1972 Myers and Liu showed that cl(G)rn/(n-2m/n). Here we show that cl(G)? n/[n (2m/n)(l + ~$1’21, where cV is the vertex degree coefficient of variation in G, that cl(G) 2 Bondy’s constructive lower bound for x(G), and that cl(G) z n/(n W(G)), where W(G) is the largest positive integer such that W(G) sd( W(G) + 1) [W(G) + 1 is the Welsh and Powell upper bound for x(G)]. It is also shown that cl(G)+f>n/(n-L(G))rn/(n-II) and that x(G)? n/(n-At); It is the largest eigenvalue of A, the adjacency matrix of G, and L(G) is a newly defined single-valued function of G similar to W(G) in its construction from the vertex degree sequence of G [L(G) 2 W(G)]. Finally, a ‘greedy’ lower bound for cl(G) is constructed from A and it is shown that this lower bound is never less than Bondy’s bound; thereafter, for 150 random graphs with varying numbers of vertices and edge densities, the values of lower bounds given in this paper are listed, thereby illustrating that this last greedily obtained lower bound is almost always better than each of the other bounds.