A not 3-choosable planar graph without 3-cycles

Every planar graph without cycles of lengths 4 to 12 is acyclically 3-choosable

An acyclic coloring of a graph G is a coloring of its vertices such that : (i) no two adjacent vertices in G receive the same color and (ii) no bicolored cycles exist in G. A list assignment of G is a function L that assigns to each vertex v ∈ V (G) a list L(v) of available colors. Let G be a graph and L be a list assignment of G. The graph G is acyclically L-list colorable if there exists an a...

Planar graphs without 3-, 7-, and 8-cycles are 3-choosable

A graph G is k-choosable if every vertex of G can be properly colored whenever every vertex has a list of at least k available colors. Grötzsch’s theorem states that every planar triangle-free graph is 3-colorable. However, Voigt [13] gave an example of such a graph that is not 3-choosable, thus Grötzsch’s theorem does not generalize naturally to choosability. We prove that every planar triangl...

Planar Graphs That Have No Short Cycles with a Chord Are 3-choosable

In this paper we prove that every planar graph G is 3-choosable if it contains no cycle of length at most 10 with a chord. This generalizes a result obtained by Borodin [J. Graph Theory 21(1996) 183-186] and Sanders and Zhao [Graphs Combin. 11(1995) 91-94], which says that every planar graph G without k-cycles for all 4 ≤ k ≤ 9 is 3-colorable.

Planar graph colorings without short monochromatic cycles

2 01 0 Every planar graph without adjacent short cycles is 3 - colorable

Two cycles are adjacent if they have an edge in common. Suppose that G is a planar graph, for any two adjacent cycles C1 and C2, we have |C1| + |C2| ≥ 11, in particular, when |C1| = 5, |C2| ≥ 7. We show that the graph G is 3-colorable.