Miniconference on the Mathematics of Computation AM8002 Discrete Mathematics and its Applications Lecture 10 – Cop Number and Genus Dr. Anthony Bonato Ryerson University

Planar graphs a graph is planar if it can be drawn in the plane without edge crossings

Facts on planar graphs let G be a planar graph with order n and size e e ≤ 3n – 6 if G has no triangles, then e ≤ 2n – 6 every planar graph has a vertex of degree at most 5

11.0 Exercise Explain why K5 is not planar. Prove that a planar graph has a proper coloring with 6 colors. (Hint: Proof is one line!)

Cops on planar graphs Theorem 11.1 (Aigner, Fromme,84): Planar graphs have cop number at most 3.

11.1 Exercise: Buckeyball graphs Prove these two graphs have cop number 3.

Ideas behind proof cop territory: induced subgraph so that if the robber entered he would eventually be caught (not necessarily immediately) cop territory starts as a maximum order isometric path inductively grow cop territory, until it is entire graph show that the cop territory is always one of three kinds, and that we can always enlarge it so it remains one of the three kinds

Outerplanar graphs a graph is outerplanar if its vertices can be arranged on a circle with the following properties: Every edge joins two consecutive vertices on the circle, or forms a chord on the circle. If two chords intersect, then they do so at a vertex.

Examples cycles each of these are outerplanar:

Cop number of outerplanar graphs Theorem 11.2 (Clarke, 02) If G is outerplanar, then it’s cop number is at most 2. proof is simpler than planar case, but still needs some care overall idea is similar: enlarge cop territory two cases: no cut vertices, or some cut vertices

Graphs on surfaces ? ? S0 S1

Genus of a graph a graph that can be embedded in an (orientable!) surface with g holes (and no fewer) has genus g planar: genus 0 toroidal: genus 1 Sg

Higher genus Schroeder’s Conjecture: If G has genus g, then c(G) ≤ g + 3. true for g = 0 (Schroeder, 01): true for g = 1 (toroidal graphs) (Quilliot,85): c(G) ≤ 2g + 3. (Schroeder,01): c(G) ≤ 3g/2 + 3 use 1 or 2 cops to reduce genus of robber territory, then use induction Cops and Robbers