Third case of the Cyclic Coloring Conjecture

On the cyclic coloring conjecture

“…Both these bounds are tight. Hebdige and Král' proved this conjecture for ∆ * (G) = 6, see [16]. So in this note we shall study only the cases when ∆ * (G) ≥ 5.…”

Cyclic coloring of plane graphs with maximum face size 16 and 17

“…Let uv be an edge such that both faces containing the edge uv are (≥ 12)-faces and u is not a 3-vertex contained in a 3-face. If u is a (≤ 4)-vertex that is contained in exactly one (≤ 4)-face f , then f sends through heavy = 17 80 to e. If u is a 4-vertex contained two 4-faces or it is a 5-face contained in two 3-faces, one each of the two faces send 1 2 through heavy = 17 160 to e. Otherwise, u sends through heavy = 17 80 to e. If v is a 3-vertex contained in a 3-face vv ′ v ′′ and the other face f ′ containing the edge v ′ v ′′ is a (≤ 11)-face, then e sends through heavy = 17 80 to f . Let uv be an edge such that one of the faces containing uv has size between 5 and 10 (inclusively).…”

“…Let uv be an edge such that one of the faces containing uv has size between 5 and 10 (inclusively). Let f be one of the face containing uv and u ′ the neighbor of u incident with f that is different from v. If the edge uu ′ is contained in a 3-face f ′ , and either u is a (≥ 6)vertex or u is a 5-vertex contained in only one 3-face, then u sends through heavy = 17 80 to e, which then sends through heavy = 17 80 to f ′ . Note that if u is a 6-vertex, then this rule may apply twice, once for each face containing the edge uv.…”

“…Finally, let uv be an edge contained in a face of size between 5 and 10 (inclusively) and a (≥ 12)-face f . Let u ′ be the neighbor of u incident with f that is different from v. If the edge uu ′ is contained in a 3-face f ′ and u is a 5-vertex contained in two 3-faces, then f ′ sends through heavy = 17 80 to e, which then sends through heavy = 17 80 to the 3-face containing u that is different from f ′ .…”

“…Despite a significant amount of interest (see e.g. [5,8,16,26,27]), the Cyclic Coloring Conjecture has been proven only for three values of ∆ ⋆ : the case ∆ ⋆ = 3, which is equivalent to the Four Color Theorem proven in [2,3] (a simplified proof was given in [25]), the case ∆ ⋆ = 4 known as Borodin's Six Color Theorem [4,6], and the recently proven case ∆ ⋆ = 6 [17]. Amini, Esperet and van den Heuvel [1], building on the work in [13,14], proved an asymptotic version of the Cyclic Coloring Conjecture: for every ε > 0, there exists ∆ 0 such that every plane graph with maximum face size ∆ ⋆ ≥ ∆ 0 has a cyclic coloring with at most 3 2 + ε ∆ ⋆ colors.…”