Toughness and binding number

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.

Toughness and edge-toughness

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.

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...

The Binding Number of a Graph and Its Anderson Number*

The binding number of a graph G, bind(G), is defined; some examples of its calculation are given, and some upper bounds for it are proved. It is then proved that, if bind(G) > c, then G contains at least 1 G 1 c/(c + 1) disjoint edges if 0 < c < 3, at least ( G I (3c 2)/3c 2(c 1)/c disjoint edges if 1 < c < $, a Hamiltonian circuit if c > $, and a circuit of length at least 3(1 G / l)(c 1)/c if...