Discrete Mathematics Letters www.dmlett.com Discrete Math. Lett. 12 (2023) 86–92 DOI: 10.47443/dml.2023.089 Research Article Optimal t-rubbling on complete graphs and paths N´ andor Sieben * Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, Arizona, USA (Received: 8 May 2023. Received in revised form: 29 June 2023. Accepted: 30 June 2023. Published online: 4 July 2023.) c  2023 the author. This is an open-access article under the CC BY (International 4.0) license (www.creativecommons.org/licenses/by/4.0/). Abstract Given a distribution of pebbles on the vertices of a graph, a rubbling move places one pebble at a vertex and removes a pebble each at two not necessarily distinct adjacent vertices. One pebble is the cost of transportation. A vertex is t-reachable if at least t pebbles can be moved to the vertex using rubbling moves. The optimal t-rubbling number of a graph is the minimum number of pebbles in a pebble distribution that makes every vertex t-reachable. The optimal t-rubbling numbers of complete graphs and paths are determined. Keywords: optimal t-rubbling; pebbling. 2020 Mathematics Subject Classification: 05C99, 05C35. 1. Introduction Graph pebbling is a simple model for the transportation of perishable resources. Let G be a connected simple graph with vertex set V .A pebble distribution p : V →{0, 1, 2,...} on G is a placement of some pebbles at the vertices of G.A pebbling move (v  u) removes two pebbles from v and places one pebble at the adjacent vertex u. We think of the lost pebble as the cost of transportation along the edge vu. A vertex r is t-reachable from a pebble distribution if at least t pebbles can be moved to r by a sequence of moves. A pebble distribution is t-solvable if every vertex is t-reachable. A recent guide to the extensive literature of graph pebbling can be found in [13]. Another useful reference is [12]. The t-pebbling number of G is the smallest number π t (G) of pebbles in a pebble distribution that forces the pebble distribution to be t-solvable. The optimal t-pebbling number of G is the least number π * t (G) of pebbles we need to create a t-solvable pebble distribution. Deciding whether π * 1 (G) ≤ k is an NP-complete problem [18]. The t-pebbling number of some graph families has been found [6, 8, 11, 16, 17]. Some optimal t-pebbling numbers were determined in [10, 19–21]. Graph rubbling allows for an extra move. A strict rubbling move (v,w  u) removes one pebble each from the distinct vertices v and w and places one pebble at the common neighbor vertex u. This time the pebbles are moved along the edges vu and wu and this transportation costs one pebble. A rubbling move is a pebbling or a strict rubbling move. Graph rubbling was introduced in [4] and further developed in [1–3, 7, 9, 14, 15]. The t-rubbling number of G is the smallest number ρ t (G) of pebbles in a pebble distribution that forces the pebble distribution to be t-solvable. The optimal t-rubbling number is the least number ρ * t (G) of pebbles we need to create a t-solvable pebble distribution. In this paper we determine the optimal t-rubbling numbers of complete graphs and paths. 2. Preliminaries We start with some basic results about graph rubbling. If the total number of pebbles on the vertices that are adjacent to a vertex v is a, then the maximum number of pebbles we can transfer to v using only these pebbles is  a 2  . This is because the pebbles can be paired up and used in rubbling moves until we run out of pebbles. Transferring pebbles between the vertices adjacent to v, instead of directly moving them to v, has no benefit. Since the expression  a 2  plays an important role in our calculations, we collect some tools that help handling it. Let pty(k) be the parity of the integer k. That is, pty(k) := 0 if k is even and pty(a) := 1 if a is odd. Then  1 2 a  = 1 2 (a - pty(a)). For x ∈ R and a ∈ Z we often use the identities d-xe = -bxc, ba + xc = a + bxc, da + xe = a + dxe. * E-mail address: nandor.sieben@nau.edu