Toughness Bounds for Fractional Hamiltonicity and Resistance Positivity

Abstract

A graph is fractionally Hamiltonian if it admits a nonnegative edge weighting in $[0,1]$ of total weight equal to its order such that every nontrivial edge cut has weight at least two. Motivated by Chvátal's Toughness Conjecture, Scheinerman and Ullman conjectured that every $2$-tough graph is fractionally Hamiltonian. In this paper, we show that every connected graph on at least three vertices that is not fractionally Hamiltonian has a non-Hamiltonian chordal spanning supergraph. Since adding edges does not decrease toughness, a theorem of Kabela and Kaiser that every $10$-tough chordal graph on at least three vertices is Hamiltonian yields that every $10$-tough graph on at least three vertices is fractionally Hamiltonian. We apply this result to resistance curvature. We prove that every fractionally Hamiltonian graph is resistance positive (RP), and consequently every $10$-tough graph is RP, confirming a conjecture of Devriendt. In the other direction, for every $\varepsilon>0$, we construct a graph that is not resistance nonnegative and has toughness greater than $3/2-\varepsilon$, extending a recent construction of Agrahari, Bibby, Boros, Garcia, Heidercheidt, and Wang.