On The Linear 2-Arboricity Of Planar Graphs Without Intersecting Short Cycles

Graph coloring theory is a hot research topic in graph theory. The linear 2-arboricity problem of planar graph is studied in this paper, this problem has important significance on coloring and decomposition of planar graph.Let G(V,E) be a simple planar graph with maximum degree △. The linear 2-arboricity of graph G is the least integer k such that G can be partitioned into k edge disjoint forests, whose component trees are paths of length at most 2. It is denoted by la2(G).Two cycles are intersecting if they have at least one common vertex. Two cycles are adjacent if they have at least one common edge.The upper bounds for the linear 2-arboricity of planar graphs with-out intersecting short cycles are given in this paper:(1) Let G be a planar graph, if G has no intersecting 3-cycles or no intersecting 4-cycles, then(2) Let G be a planar graph, if G has no intersecting 3-cycles and no intersecting 4-cycles, then(3) Let G be a planar graph, if any 3-cycles in not intersect with any 5-cycles, thenAn upper bound for the linear 2-arboricity of planar graphs without adjacent short cycles is obtained. Let graph G be a planar graph. If any i—cycle is not adjacent to a j-ycle for i,j ∈{3,4}, then...