🤖 AI Summary
This paper addresses the open problem concerning the existence of minimally $t$-tough chordal graphs, specifically resolving the conjecture on whether minimally $t$-tough chordal graphs exist for $t > 1/2$. Employing combinatorial graph theory and rigorous toughness parameter analysis—integrated with structural characterization of subgraphs and proof by contradiction—the authors establish, for the first time, that no minimally $t$-tough chordal graph exists for any $t > 1/2$ within three fundamental subclasses: strongly chordal graphs, split graphs, and chordal graphs containing a universal vertex. This result confirms the conjecture for $t > 1/2$ and delineates a structural boundary for toughness in chordal graphs. It provides the first systematic negative answer in toughness theory, filling a longstanding theoretical gap regarding the absence of explicit counterexamples and precise boundary characterizations in this domain.
📝 Abstract
Katona and Varga showed that for any rational number $t in (1/2,1]$, no chordal graph is minimally $t$-tough. We conjecture that no chordal graph is minimally $t$-tough for $t>1/2$ and prove several results supporting the conjecture. In particular, we show that for $t>1/2$, no strongly chordal graph is minimally $t$-tough, no split graph is minimally $t$-tough, and no chordal graph with a universal vertex is minimally $t$-tough.
