Geodesic Distance in Planar Graphs: An Integrable Approach

Geodesic Distance in Planar Graphs: An Integrable Approach

We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey non-linear recursion relations on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of the KP hierarchy. We obtain ...

Geodesic Embeddings and Planar Graphs

Manhattan-Geodesic Embedding of Planar Graphs

In this paper, we explore a new convention for drawing graphs, the (Manhattan-) geodesic drawing convention. It requires that edges are drawn as interior-disjoint monotone chains of axis-parallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1-bend embeddability on the grid. For the latter quest...

A unified approach to distance-two colouring of planar graphs

We introduce the notion of (A,B)-colouring of a graph: For given vertex sets A, B, this is a colouring of the vertices in B so that both adjacent vertices and vertices with a common neighbour in A receive different colours. This concept generalises the notion of colouring the square of graphs and of cyclic colouring of plane graphs. We prove a general result which implies asymptotic versions of...

Toric Integrable Geodesic Flows

By studying completely integrable torus actions on contact manifolds we prove a conjecture of Toth and Zelditch that toric integrable geodesic flows on tori must have flat metrics.