Treewidth and gonality of glued grid graphs

Abstract

We compute the treewidth of a family of graphs we refer to as the glued grids, consisting of the stacked prism graphs and the toroidal grids. Our main technique is constructing strict brambles of large orders. We discuss connections to divisorial graph theory coming from tropical geometry, and use our results to compute the divisorial gonality of these graphs.

On the Gonality of Cartesian Products of Graphs

“…Plugging in u 2 = 2 yields 2n + 1, and plugging in u 2 = n − 2 yields 5n − 11. Both of these are larger than 2n − 2 for n 4, and one of them is the minimum value of −u 2 2 + (n + 3)u 2 − 1 for 2 u n − 2; it follows that u 2 = 1. Call the one unburned vertex in the second row v 2 .…”

“…• The stacked prism graph Y m,n is the product C m P n . By [2] for m = 2n and in general by [17], this graph is known to have gonality min{m, 2n} [2], which is equal to min{|V (C m )| gon(P n ), |V (P n )| gon(C m )}.…”

“…• The toroidal grid graph T m,n is the product C m C n of two cycle graphs. By [2] for |m − n| 2 and in general by [17], the graph T m,n has gonality 2 min{m, n} [2], which is equal to min{|V (C m…”

“…Each unburned vertex has n − u 2 + 3 burning edges incident to it, except possibly for one in the same column as v 1 , which would have n − u 2 + 2 burning edges. This means there are at least (u 2 − 1)(n − u 2 + 3) + (n − u 2 + 2) = −u 2 2 + (n + 3)u 2 − 1 chips on this row. Plugging in u 2 = 2 yields 2n + 1, and plugging in u 2 = n − 2 yields 5n − 11.…”

“…In this paper, we study the gonality of the Cartesian product G H of two graphs G and H. This is the first such systematic treatment for these types of graphs, although many conjectures have been posed on the gonality of particular products [2,24,25]. Our main result is that if G and H have at least two vertices each, then G H satisfies Conjecture 1.…”