Bordeaux 3-color conjecture and 3-choosability

Bordeaux 3-color conjecture and 3-choosability

A graph G = (V ,E) is list L-colorable if for a given list assignment L = {L(v) : v ∈ V }, there exists a proper coloring c of G such that c(v) ∈ L(v) for all v ∈ V . If G is list L-colorable for every list assignment with |L(v)| k for all v ∈ V , then G is said to be k-choosable. In this paper, we prove that (1) every planar graph either without 4and 5-cycles, and without triangles at distance...

A relaxation of the Bordeaux Conjecture

A (c1, c2, ..., ck)-coloring of G is a mapping φ : V (G) 7→ {1, 2, ..., k} such that for every i, 1 ≤ i ≤ k, G[Vi] has maximum degree at most ci, where G[Vi] denotes the subgraph induced by the vertices colored i. Borodin and Raspaud conjecture that every planar graph without intersecting triangles and 5-cycles is 3-colorable. We prove in this paper that every planar graph without intersecting ...

Edge-choosability and total-choosability of planar graphs with no adjacent 3-cycles

Let G be a planar graph with no two 3-cycles sharing an edge. We show that if ∆(G) ≥ 9, then χ′l(G) = ∆(G) and χ ′′ l (G) = ∆(G) + 1. We also show that if ∆(G) ≥ 6, then χ ′ l(G) ≤ ∆(G) + 1 and if ∆(G) ≥ 7, then χ′′ l (G) ≤ ∆(G) + 2. All of these results extend to graphs in the projective plane and when ∆(G) ≥ 7 the results also extend to graphs in the torus and Klein bottle. This second edge-c...

On 3-choosability of plane graphs having no 3-, 6-, 7- and 8-cycles

A graph is k-choosable if it can be colored whenever every vertex has a list of available colors of size at least k. It is a generalization of graph coloring where all vertices do not have the same available colors. We show that every triangle-free plane graph without 6-, 7-, and 8-cycles is 3-choosable.

On 3-choosability of plane graphs without 3-, 8- and 9-cycles