Isolated toughness and path-factor uniform graphs

Normalized Tenacity and Normalized Toughness of Graphs

In this paper, we introduce the novel parameters indicating Normalized Tenacity ($T_N$) and Normalized Toughness ($t_N$) by a modification on existing Tenacity and Toughness parameters.  Using these new parameters enables the graphs with different orders be comparable with each other regarding their vulnerabilities. These parameters are reviewed and discussed for some special graphs as well.

Isolated Toughness and Existence of [a, b]-factors in Graphs

For a graph G with vertex set V (G) and edge set E(G), let i(G) be the number of isolated vertices in G, the isolated toughness of G is defined as I(G) = min{|S|/i(G− S) | S ⊆ V (G), i(G− S) ≥ 2}, if G is not complete; and I(Kn) = n − 1. In this paper, we investigate the existence of [a, b]-factor in terms of this graph invariant. We proved that if G is a graph with δ(G) ≥ a and I(G) ≥ a, then ...

Toughness and Triangle-Free Graphs

On the Toughness of Graphs

The toughness of a graph is the graph’s vulnerability to having vertices removed. For example, if vertices of the graph represent homes, and the edge of the graph represent telephone lines connecting them, then the toughness measures how badly telephone communication can broken down by relatively few lightening strikes. In this paper the maximum networks are obtained with prescribed order and t...

The toughness of cubic graphs

The toughness of a graph G is the minimum of |S|/k(G − S) over all sets S of vertices such that k(G − S) ≥ 2. In this paper upper bounds on the toughness of a cubic graph are derived in terms of the independence number and coloring parameters. These are applied to cycle permutation graphs. Running head: as per title.