9 research outputs found
Pancyclicity of highly connected graphs
A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a
graph satisfies , where is the
vertex-connectivity of , then has a Hamilton cycle. We prove a similar
result implying that a graph is pancyclic, namely it contains cycles of all
lengths between and : if is large and ,
then is pancyclic. This confirms a conjecture of Jackson and Ordaz (1990)
for large graphs, and improves upon a very recent result of Dragani\'c,
Munh\'a-Correia, and Sudakov.Comment: 31 pages, 11 figure
Toughness and Triangle-Free Graphs
In this paper, we prove that there exist triangle-free graphs with arbitrarily large toughness, thereby settling a longstanding open question. We also explore the problem of whether there exists a t-tough, n/(t + 1)-regular, triangle-free graph on n vertices for various values of t, and provide a relatively complete answer for small values of t