Edge Coloring Of Planar Graphs

Let G = (V, E) be a finite, simple and undirected graph with the set of vertices V and the set of edges E, and C = {1, 2, …, k}, a set of k colors. A proper k-edge-coloring of G is a mapping from E into C such that no adjacent edges receive the same color. If G has a proper k-edge-coloring, then G is said to be A;-edge-colorable.The chromatic index, denoted by x'(G), is the least non-negative integer k such that G is k-edge-colorable.An edge-list-assignment L of G is a set of lists of colors which assign each edge e of G a list of colors L(e). If G has a proper edge-coloring φ such that φ(e) ∈ L(e) for every e ∈ E, then G is said to be L-colorable. If G is L-colorable for every edge-list-assignment L with |L(e)| = k for every e ∈ E, then we say that G is k-list-edge-colorable, or k-edge-choosable. The edge choice number, or, list-chromatic index of G, denoted by (?)(G), is the least non-negative integer k such that G is k-edge-choosable.On edge-coloring of graphs, Vizing made a great breakthrough in 1964. He proved that x'(G) = △(G) or x'(G) = △+ 1. This famous theorem divides finite simple graphs into two classes according to their chromatic indices, namely, a graph is said to be of class 1 if x'(G) = △(G); of class 2, otherwise.Following precedent works, this dissertation studies classification problems of planar graphs. The main results obtained in this dissertations may be summarized as follows:(1) If G is a planar graph with maximum degree 6 and without 7-cycles, then G is of class 1.(2) Let G be a planar graph with maximum degree 5. G is of class 1...