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An l<\/jats:italic>-facial edge-coloring of a plane graph G<\/jats:italic> is a coloring of edges of G<\/jats:italic> such that any two edges, that share the same facial trail of length at most \n \n $$l + 1$$<\/jats:tex-math>\n \n \n l<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, receive distinct colors. It is an edge variant of the l<\/jats:italic>-facial vertex coloring, which arose as a generalization of the well-known cyclic coloring. It was conjectured by Lu\u017ear et al. in 2015 that every plane graph admits an l<\/jats:italic>-facial edge-coloring with at most \n \n $$3l + 1$$<\/jats:tex-math>\n \n \n 3<\/mml:mn>\n l<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> colors for any \n \n $$l \\ge 1$$<\/jats:tex-math>\n \n \n l<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. It is known that the bound \n \n $$3l+1$$<\/jats:tex-math>\n \n \n 3<\/mml:mn>\n l<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is tight for general plane graphs. The conjecture was recently confirmed for \n \n $$l \\le 3$$<\/jats:tex-math>\n \n \n l<\/mml:mi>\n \u2264<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> by Hor\u0148\u00e1k, Lu\u017ear and \u0160torgel (3-facial edge-coloring of plane graphs, Discrete Math. 346 (2023) 113312). In this note we prove that the conjecture holds, in the case when \n \n $$l \\ge 4$$<\/jats:tex-math>\n \n \n l<\/mml:mi>\n \u2265<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, for every graph whose reduction (the graph obtained from G<\/jats:italic> by suppressing all its 2-vertices) is 3-edge connected, and the length of the longest path in G<\/jats:italic> with interior vertices of degree 2 is at most \n \n $$\\frac{3l + 1}{10}$$<\/jats:tex-math>\n \n \n \n 3<\/mml:mn>\n l<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 10<\/mml:mn>\n <\/mml:mfrac>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00373-025-02904-x","type":"journal-article","created":{"date-parts":[[2025,2,26]],"date-time":"2025-02-26T20:59:28Z","timestamp":1740603568000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on the Facial Edge-Coloring Conjecture"],"prefix":"10.1007","volume":"41","author":[{"given":"Stanislav","family":"Jendrol\u2019","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4811-3955","authenticated-orcid":false,"given":"Alfr\u00e9d","family":"Onderko","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,2,26]]},"reference":[{"key":"2904_CR1","first-page":"429","volume":"21","author":"K Appel","year":"1977","unstructured":"Appel, K., Haken, W.: Every planar graph is four colorable. 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