5 research outputs found
Towards the Overfull Conjecture
Let be a simple graph with maximum degree denoted as . An
overfull subgraph of is a subgraph satisfying the condition . In 1986, Chetwynd and Hilton
proposed the Overfull Conjecture, stating that a graph with maximum degree
has chromatic index equal to if and
only if it does not contain any overfull subgraph. The Overfull Conjecture has
many implications. For example, it implies a polynomial-time algorithm for
determining the chromatic index of graphs with , and implies several longstanding conjectures in the area of
graph edge colorings. In this paper, we make the first improvement towards the
conjecture when not imposing a minimum degree condition on the graph: for any
, there exists a positive integer such
that if is a graph on vertices with , then the Overfull Conjecture holds for . The previous
best result in this direction, due to Chetwynd and Hilton from 1989, asserts
the conjecture for graphs with .Comment: arXiv admin note: text overlap with arXiv:2205.08564,
arXiv:2105.0528