Carl Yerger

Five-coloring graphs on the Klein bottle

We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list.

Pebbling Graphs of Diameter Three and Four

Abstract Given a configuration of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex, and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each configuration of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. We improve on the bound of Bukh by showing that the pebbling number of a graph of diameter three on n vertices is at most ⌊ 3 n / 2 ⌋ + 2 , and this bound is best possible. We obtain an asymptotically best possible bound of 3 n / 2 + Θ ( 1 ) for the pebbling number of graphs of diameter four. Finally, we prove an asymptotic upper bound for the pebbling number of graphs of diameter d, namely ( 2 ⌈ d 2 ⌉ − 1 ) n + O ( 1 ) , and this also improves a bound given by Bukh.

Threshold and complexity results for the cover pebbling game

Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, @c(G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on every vertex simultaneously, no matter how the pebbles are initially distributed. We determine Bose-Einstein and Maxwell-Boltzmann cover pebbling thresholds for the complete graph. Also, we show that the cover pebbling decision problem is NP-complete.

Optimal Pebbling on Grids

Given a distribution of pebbles on the vertices of a connected graph G, a pebbling move is defined as the removal of two pebbles from some vertex and the placement of one of these on an adjacent vertex. The pebbling number of a graph G is the smallest integer k such that for each vertex v and each distribution of k pebbles on G there is a sequence of pebbling moves that places at least one pebble on v. We say such a distribution is solvable. The optimal pebbling number of G, denoted

Six-Critical Graphs on the Klein Bottle

varPi _{OPT}(G)

Modified linear programming and class 0 bounds for graph pebbling

ΠOPT(G), is the least k such that some particular distribution of k pebbles is solvable. In this paper, we strengthen a result of Bunde et al. relating to the optimal pebbling number of the 2 by n square grid by describing all possible optimal configurations. We find the optimal pebbling number for the 3 by n grid and related structures. Finally, we give a bound for the analogue of this question for the infinite square grid.

A Boxing-like Round-by-round Analysis of American College Basketball

Let a , b ] denote the integers between a and b inclusive and, for a finite subset X ? Z , let diam ( X ) = max ( X ) - min ( X ) . We write X < p Y provided max ( X ) < min ( Y ) . For a positive integer m , let f ( m , m , m ; 2 ) be the least integer N such that any 2 -coloring Δ : 1 , N ] ? { 0 , 1 } has three monochromatic m -sets B 1 , B 2 , B 3 ? 1 , N ] (not necessarily of the same color) with B 1 < p B 2 < p B 3 and diam ( B 1 ) ? diam ( B 2 ) ? diam ( B 3 ) . Improving upon upper and lower bounds of Bialostocki, Erd?s and Lefmann, we show that f ( m , m , m ; 2 ) = 8 m - 5 + ? 2 m - 2 3 ? + ? for m ? 2 , where ? = 1 if m ? { 2 , 5 } and ? = 0 otherwise.

Cover pebbling numbers and bounds for certain families of graphs

Abstract We exhibit an explicit list of nine graphs such that a graph drawn in the Klein bottle is 5-colorable if and only if it has no subgraph isomorphic to a member of the list. This answers a question of Thomassen [J. Comb. Theory Ser. B 70 (1997), 67–100] and implies an earlier result of Kral, Mohar, Nakamoto, Pangrac and Suzuki that an Eulerian triangulation of the Klein bottle is 5-colorable if and only if it has no complete subgraph on six vertices.