Disproof of a conjecture on minimally t-tough graphs

Abstract

Let G be a graph and let ω(G) denote the number of components of G. The graph G is said to be t-tough if t⋅ω(G−X)≤|X| for all X⊆V(G) with ω(G−X)>1, where t is a nonnegative real number. The toughnessτ(G) of the graph G is the maximum value of t such that G is t-tough (taking τ(Kn)=∞ for all n≥1). A graph G is said to be minimally t-tough if τ(G)=t and τ(G−e)<t for every e∈E(G). Kriesell (2003) conjectured that every minimally 1-tough graph has a vertex of degree 2. Katona and Varga (2018) proposed a generalized version that every minimally t-tough graph has a vertex of degree ⌈2t⌉ for any positive real number t. In this note, we disprove the Generalized Kriesell's Conjecture by constructing an infinite family of counterexamples having toughness close to 1, and thus also show that the toughness condition in Kriesell's Conjecture would be sharp.