Every planar graph without adjacent cycles of length at most 8 is 3-choosable
نویسندگان
چکیده
منابع مشابه
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...
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In this paper, a structural theorem about toroidal graphs is given that strengthens a result of Borodin on plane graphs. As a consequence, it is proved that every toroidal graph without adjacent triangles is (4, 1)∗-choosable. This result is best possible in the sense that K7 is a non-(3, 1)∗-choosable toroidal graph. A linear time algorithm for producing such a coloring is presented also. © 20...
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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.
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We prove that every planar graph in which no i-cycle is adjacent to a j-cycle whenever 3 ≤ i ≤ j ≤ 7 is 3-colorable and pose some related problems on the 3-colorability of planar graphs.
متن کاملPlanar 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...
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ژورنال
عنوان ژورنال: European Journal of Combinatorics
سال: 2019
ISSN: 0195-6698
DOI: 10.1016/j.ejc.2019.07.006