Overfull graph

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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[edit]

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[edit]

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[edit]

  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 .