$(3a:a)$-List-Colorability of Embedded Graphs of Girth at Least Five

Dvořák, Zdeněk; Hu, Xiaolan

doi:10.1137/18m1214056

Cited by 1 publication

(12 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…R1 Each face other than the outer one sends 1 3 to each incident vertex that either has degree two and belongs to V (C), or has degree three and does not belong to V (C).…”

Section: Initial Charge and Discharging Rulesmentioning

confidence: 99%

“…R6 Each type-B face sends 1 3 to each tight type-A-2 face with which it shares an edge incident with x.…”

Section: R3mentioning

confidence: 99%

“…By R1, f sends at most 1 3 to each incident vertex. If f is a type-C face, then f sends 1 3 to each connected type-A-2 face by R7. Recall that f is connected to at most one type-A-2 face and f is incident with at least three 4 + -vertices, i.e., the number of vertices to that f sends charge is at most 2.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Recall that f is connected to at most one type-A-2 face and f is incident with at least three 4 + -vertices, i.e., the number of vertices to that f sends charge is at most 2. Then ch(v) ≥ ch 0 (v) − 1 3 − 2 × 1 3 = 0. If f is a type-D face, then f sends 1 3 to each connected type-A-2 face by R8.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Then ch(v) ≥ ch 0 (v) − 1 3 − 2 × 1 3 = 0. If f is a type-D face, then f sends 1 3 to each connected type-A-2 face by R8. Suppose that f is connected to p type-A-2 faces.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 4 more Smart Citations

Fractional Coloring of Planar Graphs of Girth Five

Dvořák¹,

Hu²

2020

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

A graph G is (a : b)-colorable if there exists an assignment of b-element subsets of {1, . . . , a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex x ∈ V (G), the graph G has a set coloring ϕ by subsets of {1, . . . , 6} such that |ϕ(v)| ≥ 2 for v ∈ V (G) and |ϕ(x)| = 3. As a corollary, every triangle-free planar graph on n vertices is (6n : 2n + 1)-colorable. We further use this result to prove that for every ∆, there exists a constant M∆ such that every planar graph G of girth at least five and maximum degree ∆ is (6M∆ : 2M∆ + 1)-colorable. Consequently, planar graphs of girth at least five with bounded maximum degree ∆ have fractional chromatic number at most 3 − 3 2M ∆ +1 .

show abstract

“…R1 Each face other than the outer one sends 1 3 to each incident vertex that either has degree two and belongs to V (C), or has degree three and does not belong to V (C).…”

Section: Initial Charge and Discharging Rulesmentioning

confidence: 99%

“…R6 Each type-B face sends 1 3 to each tight type-A-2 face with which it shares an edge incident with x.…”

Section: R3mentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Proof Of Theoremmentioning

confidence: 99%

“…Then ch(v) ≥ ch 0 (v) − 1 3 − 2 × 1 3 = 0. If f is a type-D face, then f sends 1 3 to each connected type-A-2 face by R8. Suppose that f is connected to p type-A-2 faces.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 3 more Smart Citations