Acyclic Coloring with Few Division Vertices
An acyclic k-coloring of a graph G is a mapping φ from the set of vertices of G to a set of k distinct colors such that no two adjacent vertices receive the same color and φ does not contain any bichromatic cycle. In this paper we prove that every triangulated plane graph with n vertices has a 1-subdivision that is acyclically 3-colorable (respectively, 4-colorable), where the number of division vertices is at most 2n − 5 (respectively, 1.5n − 3.5). On the other hand, we prove an 1.28n (respectively, 0.3n) lower bound on the number of division vertices for acyclic 3-colorings (respectively, 4-colorings) of triangulated planar graphs. Furthermore, we establish the NP-completeness of deciding acyclic 4-colorability for graphs with the maximum degree 5 and for planar graphs with the maximum degree 7.
KeywordsPlanar Graph Maximum Degree Outer Face Outer Vertex Bipartite Planar Graph
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