A new lower bound on graph gonality

On the scramble number of graphs

“…It follows by work in [23] that the set of graphs with scramble number at most k can be characterized by a finite set of forbidden immersion minors. For k = 1, the unique forbidden immersion minor is the multigraph with two vertices connected by two edges; this follows immediately from the fact that a (connected) graph has scramble number 1 if and only if it is a tree [19,Corollary 4.7]. For k = 2 we prove that there are three minimal forbidden immersion minors in Proposition 3.7.…”

“…We now present the definition of and background on scrambles, which were introduced in [19] as a generalization of brambles. A scramble S = {E 1 , .…”

“…The scramble number of a graph G is the maximum order of any scramble on G. Thus the cube graph Q 3 has sn(Q 3 ) ≥ 4; we will see later that in fact sn(Q 3 ) = 4, so there exists no scramble of higher order on Q 3 . By [19,Theorem 1.1], the scramble number of a graph is at least as large as its treewidth: tw(G) ≤ sn(G).…”

“…One of the short-comings of scramble number is that it is not quite as well-behaved as treewidth. Although scramble number is invariant under subdividing and refining edges, and is monotone under taking subgraphs, it is not minor monotone: there exists graphs G and H with H a minor of G and sn(H) > sn(G) [19,Example 4.2]. Our first main result is that scramble number is monotone under the immersion minor relation.…”

“…The most powerful lower bound known to date is the scramble number of a graph, the focus of this paper. This invariant was introduced in [19], where it was proved that the scramble number is at least as large as treewidth, and is no larger than gonality: tw(G) ≤ sn(G) ≤ gon(G).…”