Local Perspectives on Planar Colouring

Abstract

In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erdos, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this thesis, we introduce the concept of a local girth list assignment: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We state and prove a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if G is a planar graph and L is a list assignment for G such that |L(v)| ≥ 3 for all v in V(G); |L(v)| ≥ 4 for every vertex v contained in a 4-cycle; and |L(v)| ≥ 5 for every vertex v contained in a triangle, then G admits an L-colouring.

Next, we generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvorak and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this thesis, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form hyperbolic families. Analogous results for list colouring were shown by Postle and Thomas. Using results on hyperbolic families proved by Postle and Thomas, we show further that this implies that locally planar graphs are 5-correspondence colourable; and, using results of Dvorak and Kawarabayashi, that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five.

Finally we show that, in general, slightly stronger hyperbolicity theorems imply that the associated family of planar graphs have exponentially many colourings. The existence of exponentially many colourings has been studied before for list-colouring: for instance, Thomassen showed (without using hyperbolicity) that planar graphs have exponentially many 5-list colourings, and that planar graphs of girth at least five have exponentially many 3-list colourings. Using our stronger hyperbolicity theorems, we prove that planar graphs of girth at least five have exponentially many 3-correspondence colourings, and that planar graphs have exponentially many 5-correspondence colourings. This latter result proves a conjecture of Langhede and Thomassen. As correspondence colouring generalizes list colouring, our theorems also provide new, independent proofs that there are exponentially many 5-list colourings of planar graphs, and 3-list colourings of planar graphs of girth at least five.