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 Thomassé. We show an upper bound on the total Betti number in terms of the number of vertex disjoint cycles in a graph. Th...

Let R be a commutative ring with $Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $T(Gamma(R))$. It is the (undirected) graph with all elements of R as vertices, and for distinct $x, yin R$, the vertices $x$ and $y$ are adjacent if and only if $x + yinZ(R)$. We study the chromatic number and edge connectivity of this graph.

A total coloring of a graph G is a coloring of its vertices and edges such that no adjacent vertices, edges, and no incident vertices and edges obtain the same color. The concept of total coloring was introduced by V. Vizing [15] and independently by M. Behzad [3]. The total chromatic number χ (G) is the smallest number of colors needed for total coloring of G. In 1965 V. Vizing and M. Behzad c...