  The total Betti number of the independence complex of a graph is an
intriguing graph invariant. Kalai and Meshulam have raised the question on its
relation to cycles and the chromatic number of a graph, and a recent conjecture
on that theme was proved by Bonamy, Charbit and Thomasse. We show an upper
bound on the total Betti number in terms of the number of vertex disjoint
cycles in a graph. The main technique is discrete Morse theory and building
poset maps. Ramanujan graphs with arbitrary chromatic number and girth log(n)
is a classical construction. We show that any subgraph of them with less than
n^0.003 vertices have smaller total Betti number than some planar graph of the
same order, although it is part of a graph with high chromatic number.
