Overfull graph

In graph theory, an overfull graph is a graph whose size is greater than the product of its maximum degree and half of its order floored, i.e. $|E| > \Delta (G) \lfloor |V|/2 \rfloor$ where $|E|$ is the size of G, $\displaystyle\Delta(G)$ is the maximum degree of G, and $|V|$ is the order of G. The concept of an overfull subgraph, an overfull graph that is a subgraph, immediately follows. An alternate, stricter definition of an overfull subgraph S of a graph G requires $\displaystyle\Delta (G) = \Delta (S)$.

Properties

A few properties of overfull graphs:

1. Overfull graphs are of odd order.
2. Overfull graphs are class 2. That is, they require at least Δ + 1 colors in any edge coloring.
3. A graph G, with an overfull subgraph S such that $\displaystyle\Delta (G) = \Delta (S)$, is of class 2.

Overfull conjecture

In 1986, Chetwynd and Hilton posited the following conjecture that is now known as the overfull conjecture.^[1]

A graph G with $\Delta (G) \geq n/3$ is class 2 if and only if it has an overfull subgraph S such that $\displaystyle \Delta (G) = \Delta (S)$.

This conjecture, if true, would have numerous implications in graph theory, including the 1-factorization conjecture.^[2]

Algorithms

For graphs in which $\Delta\ge\frac{n}{3}$, there are at most three induced overfull subgraphs, and it is possible to find an overfull subgraph in polynomial time. When $\Delta\ge\frac{n}{2}$, there is at most one induced overfull subgraph, and it is possible to find it in linear time.^[3]

References

1. Chetwynd, A. G.; Hilton, A. J. W. (1986), "Star multigraphs with three vertices of maximum degree", Mathematical Proceedings of the Cambridge Philosophical Society 100 (2): 303–317, doi:10.1017/S030500410006610X, MR 848854.
2. Chetwynd, A. G.; Hilton, A. J. W. (1989), "1-factorizing regular graphs of high degree—an improved bound", Discrete Mathematics 75 (1-3): 103–112, doi:10.1016/0012-365X(89)90082-4, MR 1001390.
3. Niessen, Thomas (2001), "How to find overfull subgraphs in graphs with large maximum degree. II", Electronic Journal of Combinatorics 8 (1), Research Paper 7, MR 1814514.