Toughness and edge-toughness

Toughness and edge-toughness

Toughness, trees, and walks

A graph is t-tough if the number of components of G\S is at most |S|/t for every cutset S ⊆ V (G). A k-walk in a graph is a spanning closed walk using each vertex at most k times. When k = 1, a 1-walk is a hamilton cycle, and a longstanding conjecture by Chvátal is that every sufficiently tough graph has a 1-walk. When k ≥ 3, Jackson and Wormald used a result of Win to show that every sufficien...

Toughness and binding number

Let τ(G) and bind(G) be the toughness and binding number, respectively, of a graph G. Woodall observed in 1973 that τ(G) > bind(G) − 1. In this paper we obtain best possible improvements of this inequality except when (1+ √ 5)/2 < bind(G) < 2 and bind(G) has even denominator when expressed in lowest terms.

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.

On the higher-order edge toughness of a graph

Chen CC., K.M. Koh and Y.H. Peng, On the higher-order edge toughness of a graph, Discrete Mathematics 111 (1993) 113-123. For an integer c, 1 <c < 1 V(G) I1, we define the cth-order edye toughness of a graph G as The objective of this paper is to study this generalized concept of edge toughness. Besides giving the bounds and relationships of the cth-order edge toughness T,(G) of a graph G, we p...