Decompositions for edge-coloring join graphs and cobipartite graphs ?
An edge-coloring is an association of colors to the edges of a graph, in such a way that no pair of adjacent edges receive the same color. A graph G G is Class1 if it is edge-colorable with a number of colors equal to its maximum degree ?(G) ? ( G ) . To determine whether a graph G G is Class1 is NP-complete [I. Holyer, The NP-completeness of edge-coloring, SIAM J. Comput. 10 (1981) 718720]. First, we propose edge-decompositions of a graph G G with the goal of edge-coloring G G with ?(G) ? ( G ) colors. Second, we apply these decompositions for identifying new subsets of Class1 join graphs and cobipartite graphs. Third, the proposed technique is applied for proving that the chromatic index of a graph is equal to the chromatic index of its semi-core, the subgraph induced by the maximum degree vertices and their neighbors. Finally, we apply these decomposition tools to a classical result [A.J.W. Hilton, Z. Cheng, The chromatic index of a graph whose core has maximum degree 2, Discrete Math. 101 (1992) 135147] that relates the chromatic index of a graph to its core, the subgraph induced by the maximum degree vertices.
edge-coloring join graph cobipartite graph chromatic index core