Crossing number additivity over edge cuts

Abstract

We prove that if G is a graph with an minimal edge cut F of size three and G 1 , G 2 are the two (augmented) components of G − F , then the crossing number of G is equal to the sum of crossing numbers of G 1 and G 2 . Combining with known results, this implies that crossing number is additive over edge-cuts of size d for d ∈ {0, 1, 2, 3}, whereas there are counterexamples for every d ≥ 4. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.

On Degree Properties of Crossing-Critical Families of Graphs

“…Note that any degree 1 wall vertex (of a tile from S) is always identified with another degree 1 wall vertex, resulting in suppression of a degree 2 vertex in the final graph G. Hence the total number of vertices in G equals Proof. An elementary checking yields that the smallest density of a simple tile in T is 3 1 5 , achieved by T a of characteristics (5,16), and the largest is 4, achieved by T c and T d of characteristics (4,16). Lemma 7.1 combined with Theorem 6.2 implies that the average degree r of an infinite family must be in the specified interval.…”

“…Proof. An elementary checking yields that the smallest density of a simple tile in T is 3 1 5 , achieved by T a of characteristics (5,16), and the largest is 4, achieved by T c and T d of characteristics (4,16). Lemma 7.1 combined with Theorem 6.2 implies that the average degree r of an infinite family must be in the specified interval.…”

Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c ≤ 12

“…Of these, 4-regular 3-critical graphs were constructed by Pinontoan and Richter [16], and 4-regular c-critical graphs are known for every c ≥ 3, c = 4 [3]. Salazar observed that the arguments of Richter and Thomassen could be applied to average degree as well, showing that an infinite family of c-crossing-critical graphs of average degree d can exist only for d ∈ (3,6], and established their existence for d ∈ [4,6). Nonexistence of such families with d = 6 was established much later by Hernández, Salazar, and Thomas [9], who proved that, for each fixed c, there are only finitely many c-crossing-critical simple graphs of average degree at least six.…”