Open Problems #25 Toughness in Graphs at the 12th Midwest Conference on Combina- Torics, Cryptography, and Computing, Mark Elling- Ham Reported on Recent Results and Conjectures Re- Lated to Chvv Atal's Conjecture on Toughness. a Graph

In Open Problems #13, I mentioned Add am's conjecture 1] that a digraph having at least one directed cycle has a single edge whose reversal reduces the number of directed cycles, which I heard in 1991. I have recently discovered that Thomassen 21] published counterexamples to this conjecture in 1987. G is t-tough if for every cutset S V (G), the size of S is at least t times the number of components of G ? S. Since a spanning cycle must enter S every time it leaves a component of G ? S, every Hamilto-nian graph must be 1-tough. Chvv atal 10] conjectured that there is some constant t such that every t-tough graph is Hamiltonian. It has long been known that there are 2 ?-tough graphs that are not Hamiltonian, and it was thought perhaps that 2-tough graphs would be Hamiltonian. Recently, Bauer, Broersma, and Veldman 2] showed that we must raise t at least to 9=4 to get a suucient condition for Hamiltonian cycles. Jackson and Wormald 15] generalized the problem by deening a k-walk to be a spanning closed walk in which each vertex is visited at most k times. They extended the standard necessity argument about cycles to show that a graph with a k-walk must be 1=k-tough. They also gave examples showing that the toughness suucient for a k-walk must be at least (1=k))1 + 1=(2k + 1)] (although that is not the value stated in 15]). Using a result of Win 22], they proved that every 1=(k ? 2)-tough graph has a k-walk. This doesn't say much when k = 2. Elling-ham and Zha 12] proved that every 4-tough graph has a 2-walk. They needed the result of 13] that every k-tough graph has a k-factor, and they needed an extension of the result of 22]. These toughness thresholds for guaranteeing k-walks do not seem to be optimal. Jackson and Wormald posed Conjecture 1: Every 1=(k ? 1)-tough graph has a k-walk. One can also ask for a spanning subgraph more restrictive than a walk but less restrictive than a cycle. A k-trail is a spanning closed trail (each edge used at most once) in which each vertex is visited at most k times. Question 2: Does there exists a constant t 0 k such that every t 0 k-tough graph has a k-trail? Pattern-avoiding Permutations Suppose that and are permutations of m] …