Geodesic
Circular $(5,2)$-coloring of sparse graphs
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 417-426

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We prove that every triangle-free graph whose subgraphs all have average degree less than $\frac{12}5$ has a circular $(5,2)$-coloring. This includes planar and projective-planar graphs with girth at least $12$.
Keywords: triangle-free graph, circular $(k,d)$-coloring, projective-planar graph. 
@article{SEMR_2008_5_a28,
     author = {O. V. Borodin and S. G. Hartke and A. O. Ivanova and A. V. Kostochka and D. B. West},
     title = {Circular $(5,2)$-coloring of sparse graphs},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {417--426},
     publisher = {mathdoc},
     volume = {5},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2008_5_a28/}
}
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O. V. Borodin; S. G. Hartke; A. O. Ivanova; A. V. Kostochka; D. B. West. Circular $(5,2)$-coloring of sparse graphs. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 5 (2008), pp. 417-426. http://geodesic.mathdoc.fr/item/SEMR_2008_5_a28/