The 4-choosability Of Some Plane Graphs Without Adjacent Triangles

Let k be a positive integer. A graph G is called k-choosable if for given lists of k colors to each vertex of G there is a vertex coloring of G such that each vertex receives a color from its own list no matter what the lists are. In this paper, it is shown some plane graphs without adjacent triangles are 4-choosable.(1) Each plane graph contains neither adjacent triangles nor adjacent 4-faces and 3-faces is 4-choosable.(2) Each plane graph contains neither adjacent triangles nor 4-faces of distance less than three is 4-choosable.For some difficulty in proving (1) and (2), we also give two important lemmas. In the end, we finished the proof of (1) and (2) by the two lemmas.(3) Let G = (V, E, F) be a plane graph with 6 4 that contains neither adjacent triangles nor adjacent 4-faces and 3-faces. Then G contains:(i) a 4-cycle such that each of the vertex is a 4-vertex; or (ii) a 6-cycle u1u2 ???u5u1 with exactly one chord u1u3, and each of the vertex is a 4-vertex; or (iii) a -subgraph.(4) Let G = (V, E, F) be a plane graph with 4 that contains neither adjacent triangles nor 4-faces of distance less than three. Then G contains :(i) a 4-cycle such that each of the vertex is a 4-vertex; or (ii) a 6-cycle u1u2???u5u1 with exactly one chord u1u3, and each of the vertex is a 4-vertex; or(iii) a -subgraph; or(iv) a cycle u1u2 ???uku1(k 5) with exactly one chord u1u3, d(u1) = 5 and other vertices are 4-vertices; or(v) a cycle u1u2 ???u5u1u2u3 ???uku1 (k 4) with exactly two chord and u5uk d(u2) = d(vk) = 5 and other vertices are 4-vertices.