A note on choosability with defect 1 of graphs on surfaces
This note proves that every graph of Euler genus $μ$ is $\lceil 2 + \sqrt{3μ+ 3}\,\rceil$--choosable with defect 1 (that is, clustering 2). Thus, allowing defect as small as 1 reduces the choice number of surface embeddable graphs below the chromatic number of the surface. For example, the chromatic number of the family of toroidal graphs is known to be $7$. The bound above implies that toroidal graphs are $5$-choosable with defect 1. This strengthens the result of Cowen, Goddard and Jesurum (1997) who showed that toroidal graphs are $5$-colourable with defect 1.
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