Total acquisition on grids

نویسندگان

  • Lori MacDonald
  • Paul S. Wenger
  • Scott Wright
چکیده

On a weighted graph G, a total acquisition move transfers weight from a vertex u to a neighbor v provided that the weight on v is at least as much as the weight on u. Starting with all vertices having weight 1, the total acquisition number of G, denoted at(G), is the minimum number of vertices with positive weight after a sequence of total acquisition moves. In [D. Lampert and P. Slater, Congr. Numer. 109 (1995), 203–210] it is shown that at(G) ≥ ⌈ |V (G)|/2 ⌉ for all G, and P5 P5 is given as an example where this bound is not sharp. In this paper, we determine at(Pn Pm) exactly when n and m are not 5 and give nontrivial upper and lower bounds on at(Pn P5). ∗ Research supported by the Rocky Mountain Middle School Math and Science Partnership — NSF award #0412343. L. MACDONALD ET AL. /AUSTRALAS. J. COMBIN. 58 (1) (2014), 137–156 138

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عنوان ژورنال:
  • Australasian J. Combinatorics

دوره 58  شماره 

صفحات  -

تاریخ انتشار 2014