Shannon multigraph

In the mathematical discipline of graph theory, Shannon multigraphs, named after Claude Shannon by Vizing (1965), are a special type of triangle graphs, which are used in the field of edge coloring in particular.

A Shannon multigraph is multigraph with 3 vertices for which either of the following conditions holds:

• a) all 3 vertices are connected by the same number of edges.
• b) as in a) and one additional edge is added.

More precisely one speaks of Shannon multigraph Sh(n), if the three vertices are connected by $\left\lfloor \frac{n}{2} \right\rfloor$, $\left\lfloor \frac{n}{2} \right\rfloor$ and $\left\lfloor \frac{n+1}{2} \right\rfloor$ edges respectively. This multigraph has maximum degree n. Its multiplicity (the maximum number of edges in a set of edges that all have the same endpoints) is $\left\lfloor \frac{n+1}{2} \right\rfloor$.

Examples

• Shannon multigraphs
• Shannon multigraph 2.svg

  Sh(2)

• Shannon multigraph 3.svg

  Sh(3)

• Shannon multigraph 4.svg

  Sh(4)

• Shannon multigraph 5.svg

  Sh(5)

• Shannon multigraph 6.svg

  Sh(6)

• Shannon multigraph 7.svg

  Sh(7)

Edge coloring

File:Multigraph-edge-coloring.svg

This nine-edge Shannon multigraph requires nine colors in any edge coloring; its vertex degree is six and its multiplicity is three.

According to a theorem of Shannon (1949), every multigraph with maximum degree $\Delta$ has an edge coloring that uses at most $\frac32\Delta$ colors. When $\Delta$ is even, the example of the Shannon multigraph with multiplicity $\Delta/2$ shows that this bound is tight: the vertex degree is exactly $\Delta$, but each of the $\frac32\Delta$ edges is adjacent to every other edge, so it requires $\frac32\Delta$ colors in any proper edge coloring.

A version of Vizing's theorem (Vizing 1964) states that every multigraph with maximum degree $\Delta$ and multiplicity $\mu$ may be colored using at most $\Delta+\mu$ colors. Again, this bound is tight for the Shannon multigraphs.

References

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External links

• Lutz Volkmann: Graphen an allen Ecken und Kanten. Lecture notes 2006, p. 242 (German)