Some Results On Acyclic Edge Coloring Of Planar Graphs Without Adjacent Short Cycles

Let G be a toroidal graph or a planar graph, and let c be a proper edge col-oring of G. If there is no bicolored cycle in G, then c is said to be acyclic The smallest number of the colors in an acyclic edge coloring of G is called the acyclic edge chromatic number of G, denoted by χa(G).The maximum degree of G is denoted by A. In this thesis, we obtain the following results:(1) If G is a minimal graph without acyclic (Δ+2)-edge coloring, then for each vertex u of G, one of the following holds.a) u has no 2-neighbor;b)n2{u)+n3(u)≤d-4;c) u has a 2-neighbor say u1, and the other neighbor of u1 is a vertex of max-imum degree.(2) Let G be a torial graph without adjacent i- and j-cycles for any i,j ∈ {3,4,5}. Then φ(G)≤Δ+2.(3) Let G be a planar graph without adjacent i- and j-cycles for any i, j€{3,4} and without adjacent 3- and 6-cycles. Then χa’{G)≤Δ+3.