Cycle-maximal triangle-free graphs

Stephane Durocher, David S. Gunderson, Pak Ching Li, Matthew Skala

Research output: Journal Article or Conference Article in JournalJournal articleResearchpeer-review


Abstract We conjecture that the balanced complete bipartite graph K ⌊ n / 2 ⌋ , ⌈ n / 2 ⌉ contains more cycles than any other n -vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k . For k = 1 , we show that any such counterexamples have n ≤ 91 and are not homomorphic to C 5 ; and for any fixed k there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a # P -complete problem in general) in a special case used by our bounds.
Original languageEnglish
JournalDiscrete Mathematics
Issue number2
Pages (from-to)274-290
Number of pages17
Publication statusPublished - 6 Feb 2015
Externally publishedYes


  • Extremal graph theory
  • Cycle
  • Triangle-free
  • Regular graph
  • Matrix permanent
  • # P -complete


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