All longest cycles in a 2-connected partial 3-tree share a common vertex

Research output: Contribution to journalArticlepeer-review


In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. In this paper, we study the related question on longest cycles instead of longest paths (to make the question valid, we require 2-connectivity). The answer for cycles is also negative and not so much attention has been given to it. Classes for which the answer is known to be positive are dually chordal graphs and 3-trees. Perhaps the main open class for the question of both paths and cycles is the class of chordal graphs. In this paper, we show that all longest cycles intersect in graphs with treewidth at most three. This class of graphs includes several important subclasses of graphs, such as series-parallel graphs and (Formula presented.) -free chordal graphs.

Original languageEnglish
Pages (from-to)690-712
Number of pages23
JournalJournal of Graph Theory
Issue number4
StatePublished - Aug 2023


  • chordal graph
  • intersection
  • longest cycle
  • partial 3-tree
  • treewidth


Dive into the research topics of 'All longest cycles in a 2-connected partial 3-tree share a common vertex'. Together they form a unique fingerprint.

Cite this