Large Cycles in Graphs around Bondy’s and Jung’s Conjectures – Modifications, Sharpness, and Perspectives

In 1980, Bondy conjectured a common generalization (depending on λ) of some well-known degree-sum conditions for a graph ensuring the existence of Hamilton cycles for λ = 1 (Ore, 1960) and dominating cycles for λ = 2 (Bondy, 1980) as special cases. The reverse (long-cycle) version of Bondy’s conjecture was proposed in 2001 due to Jung. The importance of these two conjectures in the field is motivated by the fact that they (as starting points) give rise to all (with few exceptions) further developments through various additional extensions and limitations. In this paper, we briefly outline all known notable achievements towards solving the problem: (i) confirmation (by the author) of Bondy’s and Jung’s conjectures for some versions that are very close to the original versions; and (ii) significant improvements (by the author) of results (i), inspiring a number of improved versions of original conjectures of Bondy and Jung. Next we derive a number of modifications from improvements in (ii), which are also very close to the original versions, but do not follow directly from the Bondy’s and Young’s conjectures. Finally, all results (both old and new) are shown to be best possible in a sense based on three types of sharpness, indicating the intervals in 0 < λ < δ + 1 where the result is sharp and the intervals where the result can be further improved, where δ denotes the minimum degree.