Coloração de arestas em grafos split com grau máximo par - Cararo, Cintia Izabel (2023)
A proper edge coloring of a graph is an assignment of colors to the edges of such that adjacent edges have distinct colors. The chromatic index of a graph , denoted ′(), is the minimum number of colors for which has a proper edge coloring. Since every pair of adjacent edges must have distinct colors, ′() ≥ Δ(), where Δ() is the maximum degree of . In 1964, Vizing established that ′() ≤ Δ() + 1 for any simple graph . Graphs with ′() = Δ() are said to be Class 1, while graphs with ′() = Δ() + 1 are said to be Class 2. Despite the tight bounds for the chromatic index, determining ′() for an arbitrary graph is a difficult computational problem, known to be NP-complete. A graph is split if its vertex set can be partitioned into a clique and a stable set . In 2012, Almeida showed that to determine the chromatic index of split graphs it is sufficient to consider the cases where every vertex in has maximum degree. Considering this fact, in this master’s dissertation, we show that if the neighborhood of a subset , formed by the vertices of with degree at most Δ()/2, has at least ⌊||/2⌋ vertices, then is Class 1. In the remaining cases we characterize the subgraph-overfull split graphs.