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Small dense on-line arbitrarily partitionable graphs

Author

Listed:
  • Bednarz, Monika
  • Burkot, Agnieszka
  • Kwaśny, Jakub
  • Pawłowski, Kamil
  • Ryngier, Angelika

Abstract

A graph G=(V,E) is arbitrarily partitionable if for any sequence (n1,…,nk) that satisfies n1+…+nk=|G| it is possible to divide V into disjoint subsets V=V1∪…∪Vk such that |Vi|=ni,i=1,…,k and the subgraphs induced by all Vi are connected. In this paper we inspect an on-line version of this concept and show that for graphs of order n, 8≤n≤14, and size greater than (n−32)+6 these two concepts are equivalent. Although our result concerns only finitely many graphs, together with a recent theorem of Kalinowski [5] it implies that arbitrarily partitionable graphs of any order n and size greater than (n−32)+6 are also on-line arbitrarily partitionable. For the proof of our main result, we show some lemmas providing sufficient conditions for a graph to be traceable or Hamiltonian-connected, and they are of interest on their own.

Suggested Citation

  • Bednarz, Monika & Burkot, Agnieszka & Kwaśny, Jakub & Pawłowski, Kamil & Ryngier, Angelika, 2024. "Small dense on-line arbitrarily partitionable graphs," Applied Mathematics and Computation, Elsevier, vol. 470(C).
    • Handle: RePEc:eee:apmaco:v:470:y:2024:i:c:s0096300324000547
    DOI: 10.1016/j.amc.2024.128582
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