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. ${\displaystyle |E|>\Delta (G)\lfloor |V|/2\rfloor }$ where ${\displaystyle |E|}$ is the size of G, ${\displaystyle \displaystyle \Delta (G)}$ is the maximum degree of G, and ${\displaystyle |V|}$ 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 ${\displaystyle \displaystyle \Delta (G)=\Delta (S)}$.

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

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

Algorithms

For graphs in which ${\displaystyle \Delta \geq {\frac {n}{3}}}$, there are at most three induced overfull subgraphs, and it is possible to find an overfull subgraph in polynomial time. When ${\displaystyle \Delta \geq {\frac {n}{2}}}$, 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 0848854.
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.