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- W4387294350 abstract "We show that there exists a constant $c > 0$ such that if $G$ is a planar graph with 5-correspondence assignment $(L,M)$, then $G$ has at least $2^{ccdot v(G)}$ distinct $(L,M)$-colourings. This confirms a conjecture of Langhede and Thomassen. More broadly, we introduce a general method showing how hyperbolicity theorems for certain families of critical graphs can be used to derive lower bounds on the number of colourings of the associated class of planar graphs. Hence our main result follows from this method plus a technical theorem (that we proved in a previous paper) involving the hyperbolicity of graphs critical for $5$-correspondence colouring. We further demonstrate our method in the case of counting 3-correspondence colourings of planar graphs of girth at least five. Finally, we use these theorems to show analogous results hold in the case of counting 5-correspondence colourings of locally planar graphs, and counting 3-correspondence colourings of locally planar graphs of girth at least five." @default.
- W4387294350 created "2023-10-03" @default.
- W4387294350 creator A5067223934 @default.
- W4387294350 creator A5073051457 @default.
- W4387294350 date "2023-09-29" @default.
- W4387294350 modified "2023-10-04" @default.
- W4387294350 title "Exponentially Many Correspondence Colourings of Planar and Locally Planar Graphs" @default.
- W4387294350 doi "https://doi.org/10.48550/arxiv.2309.17291" @default.
- W4387294350 hasPublicationYear "2023" @default.
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